Asked by adaleta
Given the function notation for the explicitly defined sequence f(n)=2^n−1
, find the fifth term of the sequence.
why is 9 wrong and 16 the correct answer
, find the fifth term of the sequence.
why is 9 wrong and 16 the correct answer
Answers
Answered by
GPT-4o mini
To find the fifth term of the explicitly defined sequence given by the function \( f(n) = 2^n - 1 \), we need to evaluate the function at \( n = 5 \).
Let's compute \( f(5) \):
\[
f(5) = 2^5 - 1
\]
Calculating \( 2^5 \):
\[
2^5 = 32
\]
Now, subtracting 1:
\[
f(5) = 32 - 1 = 31
\]
Thus, the fifth term of the sequence is 31.
Now, regarding the numbers 9 and 16:
- If someone suggested that the fifth term could be 9, perhaps they were mistakenly applying different values of \( n \):
- \( f(4) = 2^4 - 1 = 16 - 1 = 15 \)
- \( f(3) = 2^3 - 1 = 8 - 1 = 7 \)
None of these values equal 9.
- For 16, they might have confused the term with \( f(4) = 15\) and thought about \( 2^4 = 16\) and made an arithmetic mistake.
However, the correct calculation for \( f(5) \) clearly shows that the fifth term is 31, not 9 or 16.
If you meant 9 and 16 in relation to somewhere else, those could perhaps be relating to earlier terms or misinterpretations, but regarding the sequence defined by \( f(n) = 2^n - 1 \), the fifth term is 31.
Let's compute \( f(5) \):
\[
f(5) = 2^5 - 1
\]
Calculating \( 2^5 \):
\[
2^5 = 32
\]
Now, subtracting 1:
\[
f(5) = 32 - 1 = 31
\]
Thus, the fifth term of the sequence is 31.
Now, regarding the numbers 9 and 16:
- If someone suggested that the fifth term could be 9, perhaps they were mistakenly applying different values of \( n \):
- \( f(4) = 2^4 - 1 = 16 - 1 = 15 \)
- \( f(3) = 2^3 - 1 = 8 - 1 = 7 \)
None of these values equal 9.
- For 16, they might have confused the term with \( f(4) = 15\) and thought about \( 2^4 = 16\) and made an arithmetic mistake.
However, the correct calculation for \( f(5) \) clearly shows that the fifth term is 31, not 9 or 16.
If you meant 9 and 16 in relation to somewhere else, those could perhaps be relating to earlier terms or misinterpretations, but regarding the sequence defined by \( f(n) = 2^n - 1 \), the fifth term is 31.
Answered by
GPT-4o mini
I apologize for the confusion in my previous response. Let me go through the sequence calculation again to clarify.
The function defined for the sequence is:
\[
f(n) = 2^n - 1
\]
To find the fifth term, we need to compute \( f(5) \):
\[
f(5) = 2^5 - 1
\]
Calculating \( 2^5 \):
\[
2^5 = 32
\]
Now subtracting 1:
\[
f(5) = 32 - 1 = 31
\]
However, if you are referring to the fourth term \( f(4) \):
\[
f(4) = 2^4 - 1
\]
Calculating \( 2^4 \):
\[
2^4 = 16
\]
Now subtracting 1:
\[
f(4) = 16 - 1 = 15
\]
### Summary:
- The fifth term \( f(5) = 31 \).
- The fourth term \( f(4) = 15 \).
If there is a misunderstanding and you mean a certain context where the output would yield 16, please clarify, but according to \( f(n) = 2^n - 1 \), the calculations stand as shown.
The function defined for the sequence is:
\[
f(n) = 2^n - 1
\]
To find the fifth term, we need to compute \( f(5) \):
\[
f(5) = 2^5 - 1
\]
Calculating \( 2^5 \):
\[
2^5 = 32
\]
Now subtracting 1:
\[
f(5) = 32 - 1 = 31
\]
However, if you are referring to the fourth term \( f(4) \):
\[
f(4) = 2^4 - 1
\]
Calculating \( 2^4 \):
\[
2^4 = 16
\]
Now subtracting 1:
\[
f(4) = 16 - 1 = 15
\]
### Summary:
- The fifth term \( f(5) = 31 \).
- The fourth term \( f(4) = 15 \).
If there is a misunderstanding and you mean a certain context where the output would yield 16, please clarify, but according to \( f(n) = 2^n - 1 \), the calculations stand as shown.
Answered by
GPT-4o mini
Thank you for your patience and clarification.
If we calculate \( f(n) = 2^{n-1} \) instead of \( f(n) = 2^n - 1 \), then the fifth term would be calculated as follows:
\[
f(5) = 2^{5-1} = 2^4
\]
Calculating \( 2^4 \):
\[
2^4 = 16
\]
So, under the revised function \( f(n) = 2^{n-1} \), the fifth term is indeed **16**.
Thank you for bringing this to my attention, and I appreciate your understanding! If you have any further questions, please feel free to ask.
If we calculate \( f(n) = 2^{n-1} \) instead of \( f(n) = 2^n - 1 \), then the fifth term would be calculated as follows:
\[
f(5) = 2^{5-1} = 2^4
\]
Calculating \( 2^4 \):
\[
2^4 = 16
\]
So, under the revised function \( f(n) = 2^{n-1} \), the fifth term is indeed **16**.
Thank you for bringing this to my attention, and I appreciate your understanding! If you have any further questions, please feel free to ask.
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