Question

Use the explicit or recursive formulas for the geometric sequence 45,135,140,1215,3645... To the sum of the first 9 terms of the sequence

Answer 1

Answer:

$S_9 = 442845$

Step-by-step explanation:

Given

$Sequence: 45,135,405,1215,3645$

Required:

The sum of the first 9 terms

We'll solve this question using the explicit formula:

Because it is a geometric sequence, we first calculate the common ratio (r).

$r = \frac{T_n}{T_{n-1}}$

Let n = 2;

So, we have:

$r = \frac{T_2}{T_{2-1}}$

$r = \frac{T_2}{T_1}$

$T_2 = 135; T_1 = 45$

So, we have:

$r = \frac{135}{45}$

$r = 3$

Explicitly, the sum of  n terms of a geometric sequence is:

$S_n = \frac{a(r^n - 1)}{r - 1}$

$a = T_1 = 45$

$r = 3$

$n = 9$

The formula becomes:

$S_n = \frac{a(r^n - 1)}{r - 1}$

$S_9 = \frac{45* (3^9 - 1)}{3 - 1}$

$S_9 = \frac{45* (3^9 - 1)}{2}$

$S_9 = \frac{45* (19683 - 1)}{2}$

$S_9 = \frac{45* (19682)}{2}$

$S_9 = 45* 9841$

$S_9 = 442845$

Hence, the sum of the first 9 terms is 442845