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Given the functions [tex]\( f(n) = 11 \)[/tex] and [tex]\( g(n) = \left(\frac{3}{4}\right)^{n-1} \)[/tex], combine them to create a geometric sequence [tex]\( a_n \)[/tex], and solve for the 9th term.

A. [tex]\( a_n = \left(11 \cdot \frac{3}{4}\right)^{n-1} ; a_9 \approx 24.301 \)[/tex]

B. [tex]\( a_n = 11\left(\frac{3}{4}\right)^{n-1} ; a_9 \approx 1.101 \)[/tex]

C. [tex]\( a_n = 11 + \left(\frac{3}{4}\right)^{n-1} ; a_9 = 11.100 \)[/tex]

D. [tex]\( a_n = 11 - \left(\frac{3}{4}\right)^{n-1} ; a_9 = 9.900 \)[/tex]



Answer :

GinnyAnswer
To solve for the 9th term in the given sequence, let's go through the steps systematically.

Given:
1. [tex]\( f(n) = 11 \)[/tex]
2. [tex]\( g(n) = \left( \frac{3}{4} \right)^{n-1} \)[/tex]

We are asked to combine these to create a geometric sequence [tex]\( a_n \)[/tex].

Option two seems to define the sequence correctly in the form:
[tex]\[ a_n = 11 \left( \frac{3}{4} \right)^{n-1} \][/tex]

Let's use this form to find [tex]\( a_9 \)[/tex].

### Step-by-Step Calculation:

1. Identify [tex]\( n \)[/tex]:
[tex]\[ n = 9 \][/tex]

2. Substitute [tex]\( n \)[/tex] into [tex]\( a_n \)[/tex]:
[tex]\[ a_9 = 11 \left( \frac{3}{4} \right)^{9-1} \][/tex]

3. Simplify the exponent:
[tex]\[ a_9 = 11 \left( \frac{3}{4} \right)^8 \][/tex]

4. Compute [tex]\( \left( \frac{3}{4} \right)^8 \)[/tex]:

[tex]\[ \left( \frac{3}{4} \right)^8 \approx 0.100113849 (rounded) \][/tex]

5. Multiply by 11:
[tex]\[ a_9 = 11 \times 0.100113849 \][/tex]
[tex]\[ a_9 \approx 1.1012420654296875 \][/tex]

Therefore, the 9th term [tex]\( a_9 \)[/tex] of the sequence is approximately [tex]\( 1.101 \)[/tex].

Hence, the correct answer is:

[tex]\[ \boxed{a_n = 11 \left( \frac{3}{4} \right)^{n-1}} \quad a_9 \approx 1.101 \][/tex]

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