Problem 18 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$3,15,75,375, \dots$$

Step-by-Step Solution

Answer

The formula for the general/nth term of the sequence is \(a_n = 3 * 5^{(n-1)}\). The seventh term of the sequence is 46,875.

Step 1: Identifying the common ratio

For a geometric sequence, the common ratio is found by dividing the second term by the first term. So, \(r = 15/3 = 5\). The common ratio for this sequence is 5.

Step 2: Writing the General Formula

We plug the values of the first term \(a = 3\) and the common ratio \(r = 5\) into the general formula. This gives us \(a_n = 3 * 5^{(n-1)}\). This is the formula for any nth term of the given geometric sequence.

Step 3: Finding the Seventh Term

Substitute \(n = 7\) into the formula for \(a_n\). This gives us \(a_7 = 3 * 5^{(7-1)} = 3 * 5^{6}\).

Step 4: Calculating the Seventh Term

Calculate the value to get \(a_7 = 3 * 15,625 = 46,875\).