To determine which of the given expressions is closest to the value of [tex]\( e \)[/tex], we need to evaluate each expression and then compare them to the known value of [tex]\( e \)[/tex]. Let's proceed step-by-step:
1. Calculate each expression:
- For option A: [tex]\(\left(1+\frac{1}{23}\right)^{23}\)[/tex]
- For option B: [tex]\(\left(1+\frac{1}{26}\right)^{26}\)[/tex]
- For option C: [tex]\(\left(1+\frac{1}{24}\right)^{24}\)[/tex]
- For option D: [tex]\(\left(1+\frac{1}{25}\right)^{25}\)[/tex]
We find that:
- [tex]\(\left(1+\frac{1}{23}\right)^{23} \approx 2.6614501186387796\)[/tex]
- [tex]\(\left(1+\frac{1}{26}\right)^{26} \approx 2.6677849665337465\)[/tex]
- [tex]\(\left(1+\frac{1}{24}\right)^{24} \approx 2.663731258068599\)[/tex]
- [tex]\(\left(1+\frac{1}{25}\right)^{25} \approx 2.665836331487422\)[/tex]
2. Compare these values to [tex]\( e \)[/tex]:
The known value of [tex]\( e \)[/tex] is approximately [tex]\( 2.718281828459045 \)[/tex].
3. Calculate the absolute difference between each expression and [tex]\( e \)[/tex]:
- For option A: [tex]\(|2.6614501186387796 - 2.718281828459045| \approx 0.05683170982026553\)[/tex]
- For option B: [tex]\(|2.6677849665337465 - 2.718281828459045| \approx 0.050496861925298564\)[/tex]
- For option C: [tex]\(|2.663731258068599 - 2.718281828459045| \approx 0.05455057039044631\)[/tex]
- For option D: [tex]\(|2.665836331487422 - 2.718281828459045| \approx 0.05244549697162304\)[/tex]
4. Determine which expression has the smallest difference:
- Option A: [tex]\(0.05683170982026553\)[/tex]
- Option B: [tex]\(0.050496861925298564\)[/tex]
- Option C: [tex]\(0.05455057039044631\)[/tex]
- Option D: [tex]\(0.05244549697162304\)[/tex]
The smallest difference is [tex]\(0.050496861925298564\)[/tex], which corresponds to option B.
Therefore, the expression [tex]\(\left(1+\frac{1}{26}\right)^{26}\)[/tex] is closest to the value of [tex]\( e \)[/tex]. The answer is:
B. [tex]\(\left(1+\frac{1}{26}\right)^{26}\)[/tex]
