Problem 74 - College Algebra

Question

Explain how to write terms of a sequence if the formula for the general term is given.

Step-by-Step Solution

Answer

To write terms of a sequence if the formula for the general term is given, substitute the position number into the formula for the general term. For the sequence with general term \(a_n = n^2\), the first four terms are \(1, 4, 9, 16\).

Step 1: Understand the Sequence

A sequence is a list of numbers or objects, where each number or object is called a term. In a sequence, the order of the terms is important.

Step 2: Understand the General Term

The formula for the general term, usually denoted by \(a_n\), defines each term of the sequence in terms of its position \(n\). For example, if the general term is \(a_n = n^2\), it means the \(n\)th term in the sequence is \(n\) squared.

Step 3: Generate the Terms

To generate the terms of the sequence, substitute the position number \(n\) into the formula for the general term. For example, to find the first four terms of the sequence with general term \(a_n = n^2\), substitute \(n = 1, 2, 3, 4\) into the formula, which gives \(a_1 = 1^2 = 1, a_2 = 2^2 = 4, a_3 = 3^2 = 9, a_4 = 4^2 = 16\). These are the first four terms of the sequence.