Outline Trig Eqns & Idents



PreCalculus: Sequences and Series

|Day |Topic |Homework |Grade |

|1 |Sequences |HW Sequences and Series – 1 | |

|2 |Convergence of sequences |HW Sequences and Series – 2 | |

|3 |Summation and series |HW Sequences and Series – 3 | |

|4 |Arithmetic sequences and series |HW Sequences and Series – 4 | |

|5 |Geometric sequences and series |HW Sequences and Series – 5 | |

|6 |Practice |HW Sequences and Series – 6 | |

|7 |Infinite series |HW Sequences and Series – 7 | |

|8 |Review |HW Sequences and Series – Review | |

|9 |**Test** | | |

PreCalculus HW: Sequences and Series - 1

1. A particular sequence is defined explicitly by: [pic] for n = 1, 2, 3, . . . . Write the first six terms of this sequence. (Note, although some of these terms can be reduced, that is generally not helpful.)

2. A particular sequence is defined recursively by: [pic]; [pic] for n > 1. Write the first six terms of this sequence.

3. The numbers 2, 5, 8, 11, 14, . . . represent an arithmetic sequence. If an represents the nth number in the sequence,

a. Find a recursive definition for the sequence.

b. Use the recursive definition of the sequence to write out the first four terms without simplifying ANY of the arithmetic (each term should involve only 2s and 3s). Use the pattern to find an explicit definition for the sequence.

4. The numbers 2, 6, 18, 54, 162, . . . represent a geometric sequence. If an represents the nth number in the sequence,

a. Find a recursive definition of the sequence.

b. Use the recursive definition of the sequence to write out the first four terms without simplifying ANY of the arithmetic (each term should involve only 2s and 3s). Use the pattern to find an explicit definition of the sequence.

5. Recursive sequences are easy to do on your calculator. Refer back to problem #2 and do the following:

4 ENTER (This takes care of a1.)

2Ans – 3 ENTER (This sets up the recursive part.)

Keep pressing ENTER.

a. Write the first five terms of the sequence defined by [pic].

b. Continue pushing ENTER on your calculator. Do the terms of the sequence seem to be approaching some value? What value? How many terms are required for the sequence to get to exactly that value?

6. Sequences that approach a fixed, finite value are said to converge and the number they converge to is called the limit of the sequence. All other sequences diverge. Tell if each of the following sequences converges and if so, give the limit.

a. [pic] b. [pic] c. [pic]

d. [pic]

The notation [x] means “the greatest integer ( x.” It can be found on your calculator as iPart under

the MATH Num menu

7. What is the practical difference between the statements an = an – 1 + 3 for n ( 2 and an + 1 = an + 3 for n ( 1?

PreCalculus HW: Sequences and Series - 2

1. Evaluate each limit or state that it does not exist.

a. [pic] b. [pic] c. [pic] d. [pic]

2. Determine if each sequence converges. If it does, find its limit.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] (Radian mode!) f. [pic]

3. For each infinite sequence, 1) write an explicit expression for the nth term and 2) find the limit of the sequence.

a. [pic] b. [pic]

c. 2, 6, 10, 14, 18, . . . d. [pic]

4. Assuming the series converges, find the limit.

a. [pic]; [pic] b. [pic]; [pic] c. [pic]; [pic]

5. Consider the sequence defined by [pic], [pic] (these are called nested radicals).

a. Write out the first four terms of this sequence. DO NOT SIMPLIFY. Can you see the pattern?

b. Assuming this series converges, find the exact value of its limit.

6. Consider the sequence defined by [pic], [pic] (these are called continued fractions.)

a. Write out the first four terms of this sequence. DO NOT SIMPLIFY. Can you see the pattern?

b. Assuming this series converges, find the exact value of its limit.

PreCalculus HW: Sequences and Series - 3

1. Write the first four terms of each infinite series.

a. [pic] b. [pic] c. [pic]

2. Write each of the following using sigma notation.

a. [pic] b. [pic]

c. [pic]

3. a. Find the sum of the first 1000 natural numbers.

b. Find the sum of the second 1000 natural numbers.

c. Find the difference [pic].

4. Use the properties of finite series to evaluate the following.

a.[pic] b. [pic] c. [pic] d. [pic]

5. Mortimer Moneybags agrees to give his son Marvin the following allowance: 5 cents on the first of January, 10 cents on the second, 15 cents on the third, and so on.

a. Write a summation to represent the total amount of money Marvin has received after n days.

b. How much total allowance would Marvin get in a (non-leap) year?

6. We want to find the area, A, enclosed by the graphs of y = x2, x = 1 and the x-axis

a. Refresh your memory: Write out the sum to approximate A using a right Riemann sum with four equal subintervals. Do not simplify anything; we are not interested in the answer, only the structure of the expression.

b. Write out the sum to approximate A using a right Riemann sum with n equal subintervals. Write the first three terms, then “. . .” then the last term.

c. Express your answer to part b in sigma notation. Evaluate the sum in terms of n.

d. Evaluate the function for n = 10, n = 100, n = 1000 and n = 10,000.

e. Assuming (correctly) that [pic] gives the exact area, what is the exact area?

PreCalculus HW: Sequences and Series - 4

1. Determine whether each of the following is an arithmetic sequence. If it is, find an in terms of n.

a. 10, 7, 4, 1, –2, . . . b. 1, 3, 6, 10, . . . c. [pic] d. 1, 1.1, 1.11, 1.111, . . .

2. Write the first five terms of the arithmetic sequence. Then write the nth term of the sequence as a function of n.

a. [pic], [pic] b. [pic], [pic]

3. Find a formula for [pic] as a function of n.

a. [pic], d = 4 b. [pic], d = 2x c. [pic], [pic]

4. Find an explicit formula for the nth term of the arithmetic sequence and find the sum of the first n terms.

a. 8, 20, 32, 44, n = 10 b. [pic], [pic], n = 25

5. Each row in a theater has two more seats than the row in front of it. If the first row seats 25 people and there are 20 rows in the theater, what is the total seating capacity?

6. On his first birthday, Rufus’ Uncle Wheezer gave him $25. On every succeeding birthday, Uncle Wheezer gave Rufus $10 more than he did the year before.

a. In terms of n, how much money did Uncle Wheezer give Rufus on his nth birthday?

b. In terms of n, how much total birthday money has Uncle Wheezer given Rufus up to and including his nth birthday?

7. Review problem #5 of HW – 3. If Marvin saves his allowance each day in a box under his bed, how many days will it take until Marvin has at least $10,000?

PreCalculus HW: Sequences and Series - 5

1. Determine whether each sequence could be geometric. If it is, find an in terms of n.

a. 2, 6, 24, 120, . . . b. 2, 6, 18, 54, . . . c. 12, 8, [pic], [pic]. . .

2. Write the first five terms of the sequence. Then write the nth term of the sequence as a function of n.

a. [pic], [pic] b. [pic], [pic]

3. Find the nth term of the geometric sequence.

a. [pic], r = [pic], n = 8 b. [pic], r = [pic], n = 12

4. In a geometric sequence, [pic] and [pic]. Find the formula for [pic]the value of [pic].

5. In a geometric sequence, [pic] and [pic]. Find the formula for [pic]the value of [pic].

6. Find the sum.

a. [pic] b. [pic]

7. Mortimer Moneybags has a 2000 gallon saltwater aquarium for his exotic tropical fish. Unfortunately, his son Marvin dumped too much salt in the aquarium. The aquarium contains 600 pounds of salt; 500 pounds is optimum for the fish. An angry Mortimer gives Marvin a one gallon container and tells him to remove one gallon of saltwater and replace it with one gallon of fresh water. Marvin is to continue doing this until the aquarium contains the correct amount of salt.

a. How much salt is in the tank after Marvin has made n one gallon replacements?

b. How many times will Marvin have to replace a gallon of water to get the salinity to the right level?

6. We want to find the area, A, enclosed by the graphs of y = 2x, x = 1 and the coordinate axes.

a. Refresh your memory: Write out the sum to approximate A using a right Riemann sum with four equal subintervals. Do not simplify anything; we are not interested in the answer, only the structure of the expression.

b. Write out the sum to approximate A using a right Riemann sum with n equal subintervals. Write the first three terms, then “. . .” then the last term.

c. Express your answer to part b in sigma notation. Evaluate the sum in terms of n.

d. Evaluate the sum for n 1,000,000.

PreCalculus HW: Sequences and Series - 6

1. Logs at a paper mill are stacked in a pile with 25 logs in the bottom layer, 24 logs in the next layer, and so on. The pile is 12 layers high. How many logs does it contain?

2. A theater is designed with 15 seats in the front row, 18 seats in the next row, 21 seats in the third row and so on. The theater is to have a total of 870 seats. How many rows is this?

3. Fast Eddie is running laps on a track one tenth of a mile long. He runs the first lap in 30 seconds. Each successive lap takes 3% longer than the previous one.

a. How long does it take Eddie to run the nth lap?

b. What total time does it take Eddie to run n laps?

c How long does it take Eddie to run 5 kilometers (approximately 3.1 miles)?

d. What total distance does Eddie run in an hour?

4. A new piece of factory equipment costs $36,000. This is the beginning of its first year.

a. What is it worth at the beginning of its nth year if it depreciates by $1440 per year?

b. What is it worth at the beginning of its nth year if it depreciates by 5% per year?

c. After how many years, n > 1, will the two models give values that are closest to being equal?

5. Norman Nerdling takes a job at Linear Applications, Inc. The starting salary is $32,000 per year and the company has guaranteed raises of $2400 per year for as long as Norman works for the company. Meanwhile, Desirae Dumbunny takes a job with Geometric Industries, Inc. The starting salary is $32,000 per year and the company has guaranteed raises of 5% per year for as long as Desirae works for the company. Norman and Desirae, both 22, plan to retire at age 55.

a. Write an explicit formula for Norman’s salary during his nth year at LA, Inc.

b. Write an explicit formula for Desirae’s salary during her nth year at GI, Inc.

c. Whose salary is higher in the second year? In the tenth year?

d. In what year does Desirae’s annual salary become greater than Norman’s?

e. Who makes more total money before retirement and by how much?

6. When dropped, a certain ball always returns to 80% of its original height. Suppose it was originally dropped from a height of 20 feet.

a. How high does the ball bounce on its first bounce?

b. How high does the ball bounce on its 6th bounce?

c. How high does the ball bounce on its nth bounce?

d. Disregarding the initial 20 foot drop, what total vertical distance has the ball traveled, both up and down, at the end of its nth bounce?

e. Imagine that the ball bounces an infinite number of times before coming to rest. What total vertical distance will the ball have traveled?

7. A weight is connected by two springs between two posts on a lab table (see diagram). The weight is pulled to the left and then released. It slides 24 centimeters to the right, then 21.12 cm to the left. Each successive slide is 12% shorter than the one before it.

a. How far does the weight travel on its nth slide?

b. What total distance has the weight traveled after its nth slide?

c. What is the last slide that is more than one centimeter in length? What total distance has the weight traveled after that slide?

d. Suppose the weight is allowed to slide back and forth forever. What total distance will it travel?

e. Suppose we only consider slides to the right. What total distance has the weight travel to the right after the nth such slide?

f. If the weight is allowed to slide forever, what percent of its total distance will have been to the right?

PreCalculus HW: Sequences and Series – 7

1. Tell whether each of the following infinite geometric series converges. If it does, find its sum.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

2. There is an old saying that “an elephant never forgets.” Suppose in its first year of life, a baby elephant gains some amount A of useful elephant knowledge. Each successive year, it learns 98% as much as it learned the previous year.

a. In terms of A, if the elephant lives 100 years, how much elephant knowledge will it have accrued?

b. If the elephant lives forever, what amount of elephant knowledge will it ultimately acquire?

3. A pendulum of length 50 cm is hanging around minding its own business when a PreCalc student taps it to the left. The pendulum swings 0.5 radian to the left of the vertical and then back to .4 radian to the right of the vertical. Assume that this is the first part of a geometric series. How much total distance will the pendulum bob travel before coming to a complete stop?

4. The midpoints of the sides of the square are joined to form a new square. This procedure is then repeated until there are n squares.

a. Find the sum of the areas of the n squares in terms of A, the area of the original square. Does this sequence converge? If so, to what?

b. Find the sum of the perimeters of the n squares in terms of P, the perimeter of the original square. Does this sequence converge? If so, to what?

5. Norman Nerdling throws a ball hard against the gym floor and watches as it rebounds straight up to a height of 12 feet. On each successive bounce, the ball rebounds to two-thirds the height of its previous bounce.

a. Find an expression for the height of the nth bounce.

b. Find an expression for the total vertical distance the ball has traveled at the end of its nth bounce (not counting the initial distance from Norman’s hand to the floor).

c. If the ball is allowed to bounce forever in this fashion, what total vertical distance will it travel (again not counting the initial hand to floor distance)?

PreCalculus Review: Sequences and Series

1. Determine if each sequence converges and if it does, state its limit.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

2. Determine if each series converges and if it does, find its sum.

a. [pic] b. [pic]

c. [pic] d. [pic]

3. It can be shown (in calculus) that [pic] when (1 < x ≤ 1.

a. Write out the first few terms of the series.

b. Write the (n + 1)st term of the series.

b. What is the value of the infinite sum [pic]?

4. Brittany goes shopping at the mall with her mother’s credit card. At the first store she spends an even $200. At each successive store, she spends $10 more than 60% of what she spent at the previous store.

a. Write a recursive formula for the sequence representing the amount of money Brittany spends in each store.

b. If Brittany shops forever, find the limit of the amount of money she spends in each store.

5. Becca goes bungee jumping. On her first downward fall, she travels 100 feet. She only rebounds 90 feet before falling again 81 feet. Suppose this sequence continues.

a. How much total distance (up and down) does Becca travel?

b. If Becca started out 105 feet above the ground, how high is she when she stops bouncing?

6. A length of half-inch diameter rope is coiled around itself on a flat surface. Assume each coil is approximately a circle and the innermost circle has a radius of 1 inch.

a. Find an explicit formula for the length of rope in the nth circle out from the center.

b. Find an explicit formula (without using () for the total amount of rope in the coil if the coil consists of n circles.

7. Old Uncle Albert has finally gone around the bend. One day he gives $25 to a total stranger on the street. Each succeeding day, Al gives money to another stranger. The amounts follow an arithmetic sequence. The 50th lucky stranger gets $760.

a. How much does lucky stranger number 120 get?

b. Assuming Uncle Al’s money lasts, which lucky stranger will receive exactly $4000?

c. How much total money has Al given away after n days?

d. If Al started with exactly $1,000,000, which lucky stranger gets the last of Al’s money? How much money does that stranger get? Is this more than the second to last lucky stranger?

8. Marvin, a slow learner, has once again put too much salt in Uncle Mortimer’s aquarium. Again, an irate Mortimer gives Marvin a one gallon bucket and tells him to remove one gallon of saltwater and replace it with one gallon of fresh water, repeating until the aquarium has the correct amount of salt. Hoping to speed up the process, Marvin tosses the gallon bucket and gets a bigger container. After After removing a container of salt water and adding a container of fresh water eight times, Marvin tests the water and finds the aquariumstill contains 566 pounds of salt. After fifteen more replacements, the aquarium is down to 547.2 pounds of salt.

a. How much salt was in the aquarium to start?

b. How many more replacements does Marvin have to make to get the salt quantity under 500 pounds?

9. Lydia Longstocking has just won the grand prize in the Gigajillions Lottery. She now has a choice: she can take $20,000,000 immediately, or $1,000,000 a year for 30 years with the first payment made immediately. To keep the problem manageable, we will make the ridiculous assumption that all her winnings are tax free. Because the promise of future money is not as valuable as immediate money in hand, Lydia needs to figure out the present value of an annuity of $1,000,000 a year for 30 years. To do this, she must assume that annual interest rates will average a certain value i over the next 30 years. Then each future $1,000,000 payment is discounted using that interest rate to see what it would be worth if she got it today:

The present value of the nth $1,000,000 payment is [pic].

a. What kind of sequence is this?

b. If Lydia assumes 5% annual interest, find the value of r.

c. The present value of the annuity is found by adding up the present values of all 30 payments. Find this present value. Should Lydia take the annuity or the immediate $20,000,000 dollars?

d. Will Lydia make a different decision if she assumes 2% annual interest?

10. The Koch snowflake (aka the Koch star) is a “simple” fractal curve.

1. Start with an equilateral triangle.

2. Trisect each side. On each middle segment,

construct a new equilateral triangle.

3. Repeat step 2.

4. Keep repeating.

a. Suppose the perimeter of the first “snowflake,” the triangle, is P1. Find the perimeter of the second snowflake.

b. Find the perimeter of the nth snowflake in terms of P1.

c. What happens to the perimeter of the snowflake as n ( (?

d. Suppose the area of the first snowflake is A. It can be shown that the area of the nth snowflake is [pic]. Rewrite this without the summation. (Note the upper limit of the sum.)

e. What happens to the area of the snowflake as n ( (?

Answers

HW – 1

1. [pic] 2. 4, 5, 7, 11, 19, 35

3a. a1 = 2; an = an – 1 + 3 for n > 1. b. an = 2 + 3(n – 1) for n ( 1.

4a. a1 = 2; an = 3an – 1 for n > 1. b. [pic] for n ( 1.

5a. 7, 5, 4, 3.5, 3.25 b. Sequence appears to approach 3

6a. Converges to 5 b. Converges to 0 c. Diverges to ( d. Diverges (it “cycles”)

7. There is none.

HW – 2

1a. 0 b. ( c. 3 d. 0

2a. [pic] b. Diverges (( () c. Diverges (alternates 1, 3, 1, 3, . . .)

d. [pic] e. Diverges f. [pic]

3a. [pic], [pic] b. [pic], [pic] c. [pic], diverges

d. [pic], [pic]

4a. [pic] b. [pic] c. [pic]

5a. [pic] b. [pic]

6a. [pic] b. [pic]

HW – 3

1a. [pic] b. [pic] c. [pic]

2a. [pic] b. [pic] c. [pic]

3a. 500,500 b. 1500500 c. 206770

4a. 360 b. 7650 c. 2380 d. 460 5a. [pic] b. $3339.95

6a. [pic] b. [pic]

c. [pic]

d. [pic], [pic], [pic], [pic] e. 1/3

HW – 4

1a. an = 10 – 3(n – 1) c. [pic]

2a. 3, 10, 17, 24, 31, . . . , 3 + 7(n – 1) b. 200, 175, 150, 125, 100, . . . , 200 – 25(n – 1)

3a. [pic] b. [pic] c. [pic]

4a. [pic], [pic] b. [pic], [pic] 5. 880

6a. [pic], [pic] 7. 632

HW – 5

1b. [pic] c. [pic]

2a. 16, 8, 4, 2, 1, . . . [pic] b. 4, –12, 36, –108, 324, . . . [pic]

3a. [pic] b. [pic] 4. [pic]; [pic]

5a. [pic]; [pic] 6a. 43 b. 943.01 7a. [pic] b. 365

6a. [pic] b. [pic]

c. [pic] d. 1.4426956 (Actual area is [pic].)

HW – 6

1. 234 2. 20

3a. [pic] b. [pic] c. 25 min d. 5.1+ mi.

4a. 36000 – 1440(n – 1) b. [pic] c. 11

5a. [pic] b. [pic] c. Norman’s (for both) d. 18th year

e. Desirae, by $238,840.67

6a. 16' b. 5.243' c. [pic] d. [pic] e. 160'

7a. [pic] b. [pic] c. The 25th; 191.81 cm d. 200 cm

e. [pic] f. (53.2%

HW – 7

1a. Converges to 27 c. Converges to 3 e. Converges to 300 b, d, f. Diverge

2a. 43.37A b. 50A 3. 250 cm

4a. [pic]; converges to 2A b. [pic]; converges to [pic]

5a. [pic] b. [pic] c. 72’

Review

1a. [pic] b. [pic] c. diverges d. [pic] e. diverges f. [pic]

2a. diverges b. [pic] c. [pic] d. diverges

3a. [pic] b. [pic] c. ln 2

4a. [pic]; [pic] b. [pic]

5a. 1000 ft b. 52.37 ft 6a. [pic] b. [pic]

7a. [pic] b. #266 c. [pic]

d. #364 gets $5380, $75 less than #363 (depending exactly how you interpret the problem)

8a. 575 lb b. 41 more (total of 64)

9a. Geometric b. r ( 0.95238 c. $16,141,073.58; take the $20M d. Probably

10a. [pic] b. [pic] c. P ( ( d. [pic] d. A ( [pic]

PreCalculus HW: Sequences and Series - 8

1. We wish to prove the following by induction: the sum of the first n odd numbers is n2.

1 + 3 + 5 + 7 + . . .+ (2n – 1) = n2 .

a. What is the first step?

b. What is the induction hypothesis?

c. What must be proved in the induction step?

d. Complete the proof.

Prove the following by mathematical induction.

2. [pic]. 3. [pic]

4. [pic] for n > 1 and x > 0

PreCalculus HW: Sequences and Series – 9

Prove the following by mathematical induction.

1. 1 + 5 + 9 + 13 + . . .+ (4n – 3) = n(2n – 1) 2. [pic]

3. [pic] 4. [pic]

Review

9. a. Prove by induction that [pic] for n ( 2. (Note: ( is a close relative of (. While ( means sum, ( means product.)

b. What is the final answer if this product is continued infinitely?

9. (See 4th following page.)

HW – 8

(See next page.)

HW – 9

(See page after next.)

HW – 8

1a. Prove that 1 = 12 (It does.)

b. Assume 1 + 3 + 5 + . . . + (2k – 1) = k2

c. Prove 1 + 3 + 5 + . . . + (2(k + 1) – 1) = (k + 1)2 or 1 + 3 + 5 + . . . + (2k + 1) = (k + 1)2

d. 1 + 3 + 5 + . . . + (2k – 1) = k2 (By assumption)

+ (2k + 1) = + (2k + 1) (Reflexive postulate)

1 + 3 + 5 + . . . + (2k + 1) = k2 + 2k + 1 (Addition postulate)

1 + 3 + 5 + . . . + (2k + 1) = (k + 1)2 (Factoring)

2. For n = 1, we have 1 = 21 – 1 which is true.

Assume 1 + 2 + 22 + 23 + . . . + 2k – 1 = 2k – 1

Need 1 + 2 + 22 + 23 + . . . + 2k = 2k + 1 – 1

1 + 2 + 22 + 23 + . . . + 2k – 1 = 2k – 1

2k = 2k

1 + 2 + 22 + 23 + . . . + 2k = 2(2k) – 1

1 + 2 + 22 + 23 + . . . + 2k = 2k + 1 – 1 as required.

3. For n = 1, we have [pic] which is true.

Assume [pic]

Need [pic]

[pic]

[pic]

[pic]=[pic]

= [pic] = [pic]

[pic] as required.

4. For n = 2, [pic] (Since x2 is positive. Think about it.)

Assume [pic]

Need [pic]

[pic]

[pic]

[pic]

[pic] as required. (Think about it.)

HW – 9

1. 1 = 1(2(1) – 1) (

Assume: 1 + 5 + 9 + 13 + . . .+ (4n – 3) = n(2n – 1)

Need: 1 + 5 + 9 + 13 + . . .+ (4(n + 1) – 3) = (n + 1)(2(n + 1) – 1)

or 1 + 5 + 9 + 13 + . . .+ (4n +1) = (n + 1)(2n + 1)

1 + 5 + 9 + 13 + . . .+ (4n – 3) = n(2n – 1)

4n + 1 = 4n + 1

1 + 5 + 9 + 13 + . . .+ (4n + 1) = n(2n – 1) + 4n +1 = 2n2 + 3n + 1 = (n + 1)(2n + 1) as required.

2. [pic] (

Assume: [pic]

Need: [pic]

[pic]

3n = 3n

[pic] = [pic] = [pic] = [pic]as required.

3. [pic] (

Assume: [pic]

Need: [pic]

or [pic]

[pic]

[pic]

[pic]

=[pic]

= [pic]

= [pic] as required.

4. [pic] (

Assume: [pic]

Need: [pic]

[pic]

xn = xn

[pic]

= [pic] as required.

Review

9. For n = 2: [pic] ( [pic] (

Hypothesis: [pic]

To prove: [pic]

Proof: [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic] as required.

2. A sequence is defined by the formula [pic] for n = 1, 2, 3, . . . .

a. Find the values of the first seven terms of this sequence.

b. What sequence is this? Write its recursive formula.

If Tn is the nth triangular number, prove that Tn + Tn – 1 = n2

Use the ratio test to tell if each series converges.

8. [pic] 9. [pic] 10. [pic]

Harmonic series

Reciprocals form an arithmetic series

[pic]

arithmetic mean of a and b

[pic]

geometric mean of a and b

[pic]

harmonic mean of a and b

[pic]

then

[pic]

5. [pic]

12. a – 1 is a factor of an – 1 for all n ≥ 1.

13. For the sequence a1 = 2, [pic]

a. the sequence is increasing. b. The sequence is bounded: an < 6 for all n ≥ 1

-----------------------

.5

.4

50

[pic]

3

2

1

etc.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download