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Answer :
Sure, let's analyze the given sequence and determine which function fits it step by step.
The sequence provided is: [tex]\(26, 35, 44, 53, 62, \ldots\)[/tex]
We need to find which function from the given options produces this sequence for [tex]\(n = \{1, 2, \ldots, n\}\)[/tex].
Let's evaluate each option:
### Option A: [tex]\( f(n) = 17 - 9n \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 17 - 9 \cdot 1 = 17 - 9 = 8
\][/tex]
This does not match 26. So, Option A is not correct.
### Option B: [tex]\( F(n) = 9n + 17 \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
F(1) = 9 \cdot 1 + 17 = 9 + 17 = 26
\][/tex]
Matched!
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[
F(2) = 9 \cdot 2 + 17 = 18 + 17 = 35
\][/tex]
Matched!
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[
F(3) = 9 \cdot 3 + 17 = 27 + 17 = 44
\][/tex]
Matched!
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[
F(4) = 9 \cdot 4 + 17 = 36 + 17 = 53
\][/tex]
Matched!
5. For [tex]\( n = 5 \)[/tex]:
[tex]\[
F(5) = 9 \cdot 5 + 17 = 45 + 17 = 62
\][/tex]
Matched!
All values match the given sequence. So, Option B is correct.
### Option C: [tex]\( f(n) = 9n + 26 \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 9 \cdot 1 + 26 = 9 + 26 = 35
\][/tex]
This does not match 26. So, Option C is not correct.
### Option D: [tex]\( f(n) = 26 - 9n \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 26 - 9 \cdot 1 = 26 - 9 = 17
\][/tex]
This does not match 26. So, Option D is not correct.
Therefore, the function that produces the given sequence is:
Option B: [tex]\( F(n) = 9n + 17 \)[/tex]
The sequence provided is: [tex]\(26, 35, 44, 53, 62, \ldots\)[/tex]
We need to find which function from the given options produces this sequence for [tex]\(n = \{1, 2, \ldots, n\}\)[/tex].
Let's evaluate each option:
### Option A: [tex]\( f(n) = 17 - 9n \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 17 - 9 \cdot 1 = 17 - 9 = 8
\][/tex]
This does not match 26. So, Option A is not correct.
### Option B: [tex]\( F(n) = 9n + 17 \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
F(1) = 9 \cdot 1 + 17 = 9 + 17 = 26
\][/tex]
Matched!
2. For [tex]\( n = 2 \)[/tex]:
[tex]\[
F(2) = 9 \cdot 2 + 17 = 18 + 17 = 35
\][/tex]
Matched!
3. For [tex]\( n = 3 \)[/tex]:
[tex]\[
F(3) = 9 \cdot 3 + 17 = 27 + 17 = 44
\][/tex]
Matched!
4. For [tex]\( n = 4 \)[/tex]:
[tex]\[
F(4) = 9 \cdot 4 + 17 = 36 + 17 = 53
\][/tex]
Matched!
5. For [tex]\( n = 5 \)[/tex]:
[tex]\[
F(5) = 9 \cdot 5 + 17 = 45 + 17 = 62
\][/tex]
Matched!
All values match the given sequence. So, Option B is correct.
### Option C: [tex]\( f(n) = 9n + 26 \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 9 \cdot 1 + 26 = 9 + 26 = 35
\][/tex]
This does not match 26. So, Option C is not correct.
### Option D: [tex]\( f(n) = 26 - 9n \)[/tex]
1. For [tex]\( n = 1 \)[/tex]:
[tex]\[
f(1) = 26 - 9 \cdot 1 = 26 - 9 = 17
\][/tex]
This does not match 26. So, Option D is not correct.
Therefore, the function that produces the given sequence is:
Option B: [tex]\( F(n) = 9n + 17 \)[/tex]
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