Question

A geometric sequence is defined by the equation an = (3)3 − n. What are the first three terms of the sequence and what is the common ratio, r?

Answer 1

Question:

A geometric sequence is defined by the equation an = (3)^3 − n. What are the first three terms of the sequence and what is the common ratio, r?

Answer:

The first three terms of sequence is 9, 3 , 1

The common ratio is $\frac{1}{3}$

Solution:

The geometric sequence is defined by the equation:

$a_n = (3)^{3-n}$

To find the first three terms of sequence, substitute n = 1, n = 2, n = 3

First term:

Put n = 1 in given equation

$a_1=(3)^{3-1}\\\\a_1 = 3^2\\\\a_1 = 9$

Thus first term of sequence is 9

Second term:

Put n = 2 in given equation

$a_2 = (3)^{3-2}\\\\a_2 = 3^1\\\\a_2 = 3$

Thus second term of sequence is 3

Third term:

Put n = 3 in given sequence

$a_3 = (3)^{3-3}\\\\a_3=3^0\\\\a_3 = 1$

Thus third term of sequence is 1

Thus the first three terms of sequence is 9, 3 , 1

To find common ratio:

Common ratio is found by dividing the two consecutive terms

$a_1 = 9\\\\a_2 = 3\\\\a_3 = 1$

Thus common ratio is obtained as:

$r = \frac{a_2}{a_1} = \frac{3}{9} = \frac{1}{3}$

$r = \frac{a_3}{a_2}=\frac{1}{3}$

Thus common ratio is $\frac{1}{3}$