All Hallows High School
Worksheet: Arithmetic series
Mr. Pevey Name:__________________________
An arithmetic series is the sum of the terms of an arithmetic sequence, defined as:
S n ’ a1 + a 2 + ... + an
where a1 is the first term and Sn is the sum of n terms.
Example: Given the sequence 1, 4, 7, 10, 13,…, the sum of the first six terms is: S6 ’ 1 + 4 + 7 + 10 + 13 + 16 ’ 51
The formula for the sum Sn of the first n terms of the sequence a n ’ a1 + (n − 1)d is :
1
S n ’ n (2a1 + (n − 1)d )
2
Example: Calculate the sum of the sequence 1, 2, 3,…, 100.
1
S100 ’ 100(2 1 + (100 − 1) 1) ’ 5050
2
Using the Greek symbol for S, called sigma, summation notation is often used to express an arithmetic series:
n
∑ai , where n is the term at which the sum ends, i is the term at which the sum
i ’1
begins, and ai is the values ( a1 , a 2 , ..., an ) being added.
Example: Using sigma notation, write the sum of the first 20 positive odd numbers.
The sequence 1, 3, 5,… can be described using the formula a n ’ a1 + (n − 1)d . ai ’ 1 + (i − 1)2 ’ 2i − 1
20
In sigma notation, we would write this as ∑(2i − 1) .
| | | |i ’1 | |
|The sum is: S20 |’ |1 |20(2 1 + (20 − 1) 2) ’ 400 . | |
| | |2 | | |
| | | | | |
1) Calculate the sum of the first thirteen terms of the sequence 3, 7, 11, 15, …
2) Calculate the sum of the first twelve multiples of 5.
Write each of the sums using sigma notation. 3) 3, 6, 9, 12, 15, 18.
Expand the sums and evaluate.
| |6 |
|5) |∑(i + 2) |
| |i ’1 |
Evaluate.
| |50 |
|7) |∑3i |
| |i ’1 |
4) 4, 14, 24, 34, 44.
| |5 |
|6) |∑(4i − 3) |
| |i ’1 |
| |10 |
|8) |∑(8 − 4n) |
| |n ’1 |
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