Answer:
[tex]S_9 = 442845[/tex]
Step-by-step explanation:
Given
[tex]Sequence: 45,135,405,1215,3645[/tex]
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).
[tex]r = \frac{T_n}{T_{n-1}}[/tex]
Let n = 2;
So, we have:
[tex]r = \frac{T_2}{T_{2-1}}[/tex]
[tex]r = \frac{T_2}{T_1}[/tex]
[tex]T_2 = 135; T_1 = 45[/tex]
So, we have:
[tex]r = \frac{135}{45}[/tex]
[tex]r = 3[/tex]
Explicitly, the sum of n terms of a geometric sequence is:
[tex]S_n = \frac{a(r^n - 1)}{r - 1}[/tex]
[tex]a = T_1 = 45[/tex]
[tex]r = 3[/tex]
[tex]n = 9[/tex]
The formula becomes:
[tex]S_n = \frac{a(r^n - 1)}{r - 1}[/tex]
[tex]S_9 = \frac{45* (3^9 - 1)}{3 - 1}[/tex]
[tex]S_9 = \frac{45* (3^9 - 1)}{2}[/tex]
[tex]S_9 = \frac{45* (19683 - 1)}{2}[/tex]
[tex]S_9 = \frac{45* (19682)}{2}[/tex]
[tex]S_9 = 45* 9841[/tex]
[tex]S_9 = 442845[/tex]
Hence, the sum of the first 9 terms is 442845
