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olchik [2.2K]
3 years ago
12

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
Mathematics
1 answer:
kkurt [141]3 years ago
5 0

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

<em>Hence, the sum of the first 9 terms is 442845</em>

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