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Chapter 8: Problem 6

Write the first five terms of each geometric sequence. $$a_{n}=-3 a_{n-1}, \quad a_{1}=10$$

Short Answer

Expert verified
The first five terms of the geometric sequence are: 10, -30, 90, -270, 810.

Step by step solution

01

Identifying the First Term

As given in the question, the first term (also called the initial term or \(a_1\)) of the geometric sequence is 10.
02

Using the recursive formula to find the second term

Use the recursive formula \(a_{n}=-3 a_{n-1}\) by substituting \(n=2\) and \(a_{1}=10\) to find the second term \(a_2\): \(a_2=-3*a_1=-3*10=-30\)
03

Calculating the third term

Again use the recursive formula \(a_{n}=-3 a_{n-1}\) by inputting \(n=3\) and \(a_{2}=-30\) to find the third term \(a_3\): \(a_3=-3*a_2=-3*(-30)=90\)
04

Finding the fourth term

Repeating the previous step with \(n=4\) and \(a_{3}=90\) will provide the fourth term \(a_4\): \(a_4=-3*a_3=-3*90=-270\)
05

Determining the fifth term

Finally, use the formula once more with \(n=5\) and \(a_{4}=-270\) to achieve the fifth term \(a_5\): \(a_5=-3*a_4=-3*(-270)=810\)

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