Problem 20 - Beginning and Intermediate Algebra

Question

Given the general term of each sequence, find each of the following. \(a_{n}=4 n^{2}-9\) a) the first term of the sequence b) the fourth term c) the 13 th term

Step-by-Step Solution

Answer

The first term of the sequence is -5, the fourth term is 55, and the 13th term is 667.

Step 1: Determine the Formula for the nth Term

The formula for the nth term in the sequence is provided as \(a_n = 4n^2 - 9\).

Step 2a: Find the First Term (n=1)

To find the first term, substitute n=1 into the formula: \(a_1 = 4(1)^2 - 9\). Calculate the result: \(a_1 = 4(1) - 9 = 4 - 9 = -5\) The first term in the sequence is -5.

Step 2b: Find the Fourth Term (n=4)

To find the fourth term, substitute n=4 into the formula: \(a_4 = 4(4)^2 - 9\). Calculate the result: \(a_4 = 4(16) - 9 = 64 - 9 = 55\) The fourth term in the sequence is 55.

Step 2c: Find the 13th Term (n=13)

To find the 13th term, substitute n=13 into the formula: \(a_{13} = 4(13)^2 -9\). Calculate the result: \(a_{13} = 4(169) - 9 = 676 - 9 = 667\) The 13th term in the sequence is 667.