Answer :
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]
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]
