Series and Sequences

Definition: A sequence is an ordered list of numbers. The sum of the terms of a sequence is a series.

In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant. This constant is known as the common difference. The following formula can be used to find the n-th term in an arithmetic sequence, where a₁ is the first term and d is the common difference:

The sum of an arithmetic sequence is an arithmetic series. To find this sum, we can add the first term and the last term of a sequence, divide by 2 to find the mean, and multiply by the total number of terms (n):

A geometric sequence is a sequence with a common ratio between consecutive terms. The following formula can be used to find the n-th term in a geometric sequence, where a₁ is the first term and r is the common ratio:

We can use the following formula to find the sum of a geometric series:

Example:  Consider the sequence: {1, 3, 5, 7, 9, …}.

a) What is the value of a₁₀?

b) Find the sum of the first 10 terms.

Solution:

a) The index of a₁₀ is n = 10, therefore the problem is asking for the value of the tenth term. Because this is a arithmetic sequence, we can use the following formula to find the value of a₁₀, where d is the common difference, 2:

Plug the given information into the formula to find a₁₀:

a₁₀ = 1 + (10 – 1)2

a₁₀ =  19

This tells us the tenth term of this sequence will be 19.

b) To find the sum of the first 10 terms, we can use the following formula, keeping in mind that we determined a₁₀ to be 19 in part a:

= 10/2 (1 + 19)

= 10/2 (20) = 100

This tells us the sum of the first 10 terms is 100.

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