In an arithmetic sequence, the difference between two ...
Alg 2 BC U11 Day 1 – Arithmetic and Geometric Sequences
Give the next three terms of each sequence below:
Sequence #1a: 4, 7, 10, 13, 16, ___, ___, ___ Sequence #1b: 4, 8, 16, 32 , ___, ___, ___
Sequence #2a: 5, 10, 15, 20, 25, ___, ___, ___ Sequence #2b: 2, 6, 18, 54 , ___, ___, ___
Sequence #3a: 50, 48, 46, 44, 42, ___, ___, ___ Sequence #3b: 100, 50, 25, 12.5, ___, ___, ___
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Make some observations about the sequences in the 1st column as compared to those in the 2nd column:
How would you find the 100th term of Sequence #1a?
How would you find the 100th term of Sequence #1b?
Each sequence in the first column is called an ARITHMETIC SEQUENCE because each has a common difference (d). To go from one term to the next, we add the common difference. (The common difference can be _________________ or ____________________.
In Sequence #1a, d = ______.
In Sequence #2a, d = ______.
In Sequence #3a, d = ______.
Each sequence in the second column is called a GEOMETRIC SEQUENCE because each has a common ratio (r). To go from one term to the next, we multiply by the common ratio.
In Sequence #1b, r = ______.
In Sequence #2b, r = ______.
In Sequence #3b, r = ______.
A sequence is a ______________ whose domain is _________________.
|Term # |1 |2 |3 |4 |… |n |
|Value ([pic]) |4 |7 |10 |16 |… | |
EXPLICIT formula for the nth term of an ARITHMETIC SEQUENCE:
EXPLICIT formula for the nth term of a GEOMETRIC SEQUENCE:
In this sequence: 5, 9, 13, 17, 21, 25, 29 ….
a) What is a3? _______
b) What is a6? _______
c) What is a87? _______
Write the explicit formula for each sequence. Then use the formula to find a28 for each sequence.
A) 5, 10, 15, 20, 25,… Explicit formula: ______________________________
a28 =
B) 2, 6, 18, 54, … Explicit formula: ______________________________
a28 =
C) 50, 48, 46, 44, 42, … Explicit formula: ______________________________
a28 =
The explicit formulas can be used in multiple ways:
1. In the sequence 8, 19, 30, 41, …, which term is 305?
2. A geometric sequence has a1 equal to -2 and a10 equal to -39,366. Find the value of r.
Arithmetic Mean: Geometric Mean:
Examples: Find the arithmetic mean of 5 and 36.
Find the geometric mean of 5 and 45.
Arithmetic and Geometric MeanS
Examples: Insert 3 arithmetic means between 2 and 34.
Insert 4 geometric means between 500 and [pic].
RECURSIVE FORMULA: A formula that defines a sequence by relating each term to _________________.
** A recursive formula must have at least ______________ _____________________.
Example #1: Write the first 5 terms of the following sequence, which is defined recursively:
a1=10
______ ______ ______ ______ ______
an=an-1+7
Example #2: Write the first 5 terms of the following sequence, which is defined recursively:
a1=-6
______ ______ ______ ______ ______
an+1=5an
Example #3: Write a RECURSIVE formula for the sequence below: (Remember it must have 2 parts.)
10, 8, 6, 4, 2, …
Example #4: Write a RECURSIVE formula for the sequence below: (Remember it must have 2 parts.)
[pic]
Practice Problems:
1. In the arithmetic sequence .7, .9, 1.1, 1.3, …, which term is 21.3?
2. If a1=98 and a37=83.6 in an arithmetic sequence, find the common difference d.
3. In the following sequence, find “n” if an = 686. [pic]
4. Find the arithmetic mean of 47 and 98.
5. Find the geometric mean of 58 and 9.28.
6. Find the missing terms in the following geometric sequence: …, 27, ___, ___, ___, [pic], …
7. Find the missing terms in the following arithmetic sequence: …, 27, ___, ___, ___, [pic], …
8. Find the number of terms in the sequence 7, 10, 13, ..., 55.
9. Find a15 for an arithmetic sequence where a3 = -4 and a6 = -13
10. Tom has a job at McDonalds and during his first week there, he earns $76. He does such a good job that each week, he earns $3 more than the previous week. Write a sequence that shows how much Tom earns during the first 5 weeks: Then write an explicit formula and determine how much Tom makes during the 52nd week: _____, _____, _____, _____, _____, …
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