Answer :
To solve this problem, we need to find the value of [tex]\( f(20) \)[/tex] based on the given recurrence relation and initial condition:
1. Identify the recurrence relation:
The function is defined as [tex]\( f(n) = f(n-1) + 5 \)[/tex].
2. Initial condition:
We know that [tex]\( f(1) = -2 \)[/tex].
3. Find [tex]\( f(20) \)[/tex]:
We can calculate [tex]\( f(n) \)[/tex] by iteratively applying the recurrence relation starting from [tex]\( f(1) \)[/tex].
Instead of calculating each term one by one, we can notice that the relation is an arithmetic sequence where each term is increased by 5 from the previous one. The general formula for an arithmetic sequence can be expressed as:
[tex]\[
f(n) = f(1) + (n-1) \times 5
\][/tex]
4. Substitute into the formula:
Plug in [tex]\( n = 20 \)[/tex] into the formula, using the known value of [tex]\( f(1) \)[/tex]:
[tex]\[
f(20) = -2 + (20-1) \times 5
\][/tex]
5. Perform the calculation:
[tex]\[
f(20) = -2 + 19 \times 5
\][/tex]
[tex]\[
f(20) = -2 + 95
\][/tex]
[tex]\[
f(20) = 93
\][/tex]
Therefore, the value of [tex]\( f(20) \)[/tex] is 93. So, the correct answer is option (1) 93.
1. Identify the recurrence relation:
The function is defined as [tex]\( f(n) = f(n-1) + 5 \)[/tex].
2. Initial condition:
We know that [tex]\( f(1) = -2 \)[/tex].
3. Find [tex]\( f(20) \)[/tex]:
We can calculate [tex]\( f(n) \)[/tex] by iteratively applying the recurrence relation starting from [tex]\( f(1) \)[/tex].
Instead of calculating each term one by one, we can notice that the relation is an arithmetic sequence where each term is increased by 5 from the previous one. The general formula for an arithmetic sequence can be expressed as:
[tex]\[
f(n) = f(1) + (n-1) \times 5
\][/tex]
4. Substitute into the formula:
Plug in [tex]\( n = 20 \)[/tex] into the formula, using the known value of [tex]\( f(1) \)[/tex]:
[tex]\[
f(20) = -2 + (20-1) \times 5
\][/tex]
5. Perform the calculation:
[tex]\[
f(20) = -2 + 19 \times 5
\][/tex]
[tex]\[
f(20) = -2 + 95
\][/tex]
[tex]\[
f(20) = 93
\][/tex]
Therefore, the value of [tex]\( f(20) \)[/tex] is 93. So, the correct answer is option (1) 93.
