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# Sequences and Series Formulas Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Overview of the n-th Partial Sums and the Index of Summation

There are times when we want to add the first n terms of a sequence. The sum

a 1 + a 2 + a 3 + ... + a n

is called the n th partial sum of the sequence. Its notation is S n .

S 1 = a 1

S 2 = a 1 + a 2

S 3 = a 1 + a 2 + a 3

S 4 = a 1 + a 2 + a 3 + a 4

Another common notation for the n th partial sum uses the capital Greek letter sigma, “Σ.” This notation also makes use of a n or a formula for a n . means “add the a n s beginning with a 1 and ending with a 5 .

The subscript n is called the index of summation . Other common indices are i , j , and k .

#### Examples

Write the sum.

Write the sum using summation notation.

• This is the sum of the first 20 terms in a sequence, so we will begin by writing The n th term of the sequence is , and the summation notation for this sum is

• 2 + 4 + 6 + 8 + 10 + 12
• This is the sum of the first six terms of the sequence whose n th term is a n = 2 n . The summation notation is .

• This is the sum of the first nine terms of the sequence whose n th term is . The summation notation is

There are formulas for finding the n th partial sum for special sequences. Using these formulas, we can add many terms of a sequence with little work. We will learn the formulas for the sums of two important sequences, Arithmetic Sequences Help and Geometric Sequences Help, later.

## Sequences and Series Practice Problems

#### Practice

1. Find the first four terms and the 100th term of the sequence whose n th term is

2. Find the first four terms and the 100th term of the sequence whose n th term is

3. Find the first four terms of the sequence whose n th term is and a 1 = 256.

4. Without using a recursive definition, find the n th term for the sequence

5. Without using a recursive definition, find the n th term for the sequence

6. Write the sum for .

7. Write the sum using summation notation.

#### Solutions

Practice problems for this concept can be found at: Sequence and Series Practice Test.

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