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# Sequences and Series Formulas Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction to Sequences and Series Formulas

A sequence is an ordered list of numbers. Although they list the same numbers, the sequence 1, 2, 3, 4, 5, 6, ... is different from the sequence 2, 1, 4, 3, 6, 5, .... Usually a sequence is infinite. This means that there is no last term in the sequence. A series is the sum (if it exists) of a sequence. Although a sequence can be any list of numbers, we will work with sequences that can be found from a formula. Formulas describe how to compute the nth term, a n For example, the formula a n = 2 n + 1 gives us this sequence.

#### Examples

Find the first four terms and the 50th term of the sequence.

• a n = n 2 − 10

The first term is a 1 = 1 2 − 10 = −9; the second term is a 2 = 2 2 − 10 = −6; the third term is a 3 = 3 2 − 10 = −1; the fourth term is a 4 = 4 2 − 10 = 6; and the 50th term is a 50 = 50 2 − 10 = 2490.

• a n = (−1) n

## Recursive Formula and the Fibonacci Sequence

Finding the terms of a sequence is the same function evaluation we did earlier. Sequences are special kinds of functions whose domain is the natural numbers (instead of intervals of real numbers).

### Recursive Formula

We can write the formulas for many sequences using the previous term. For example, the next term of the sequence 3, 5, 7, 9, ... can be found by adding 2 to the previous term. In other words, we could use the formula a n = a n −1 + 2. This is a recursive formula. This formula is not of much use unless we know how to start. For this reason, the value of a 1 is usually given with recursively defined sequences. A complete recursive definition for this sequence is a n = a n −1 + 2, a 1 = 3. Now we can compute the terms of the sequence.

#### Examples

Find the first four terms of the sequence.

• a n = 3 a n −1 + 5, a 1 = −4
• Think of 3 a n −1 + 5 as “3 times the previous term plus 5.”

a 1 = −4

a 2 = 3(−4) + 5 = −7

a 3 = 3(−7) + 5 = −16

a 4 = 3(−16) + 5 = −43

• The terms of this sequence are found by taking the quotient of the previous two terms.

### Fibonacci Sequence

A famous recursively defined sequence is the Fibonacci Sequence. Entire books are written about it! The n th term of the Fibonacci Sequence is a n = a n −1 + a n −2 and a 1 = 1 and a 2 = 1. From the third term on, each term is the sum of the previous two terms. The first few terms are 1, 1, 2, 3, 5, 8, 13, ....

Instead of using a formula to describe a sequence, we might be given the first few terms. From these terms we should be able to see enough of a pattern to write a formula for the n th term.

#### Examples

Find the next term in the sequence.

• 2, 6, 18, 54, ...
• The next term is 3(54) = 162.

• The next term is

• 1, −2, 4, −8, 16, ...
• The next term is −2(16) = −32.

Find a formula for the n th term for the next four examples. Do not use a recursive definition.

• 3, 9, 27, 81, ...
• 3 = 3 1 , 9 = 3 2 , 27 = 3 3 , 81 = 3 4

The n th term is a n = 3 n .

• −2, −4, −6, −8, −10, ...
• −2 = −2(1), −4 = −2(2), −6 = −2(3), −8 = −2(4), −10 = −2(5)

The n th term is a n = −2 n .

• −1, 4, −9, 16, −25, ...
• −1 = −1 2 , 4 = 2 2 , −9 = −3 2 , 16 = 4 2 , −25 = −5 2

If we want the signs to alternate, we can use the factor (−1) n (if we want the odd-numbered terms to be negative) or (−1) n +1 (if we want the even-numbered terms to be negative). The n th term of this sequence is a n = (−1) n n 2 .

The n th term is

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