To determine which of the given expressions is closest to [tex]\( e \)[/tex], we first need to evaluate each expression.
The given expressions are:
- [tex]\( A = \left(1+\frac{1}{29}\right)^{29} \)[/tex]
- [tex]\( B = \left(1+\frac{1}{30}\right)^{30} \)[/tex]
- [tex]\( C = \left(1+\frac{1}{27}\right)^{27} \)[/tex]
- [tex]\( D = \left(1+\frac{1}{28}\right)^{28} \)[/tex]
Let's evaluate each expression:
1. [tex]\( A = \left(1+\frac{1}{29}\right)^{29} \approx 2.6728491439808066 \)[/tex]
2. [tex]\( B = \left(1+\frac{1}{30}\right)^{30} \approx 2.6743187758703026 \)[/tex]
3. [tex]\( C = \left(1+\frac{1}{27}\right)^{27} \approx 2.6695939778125704 \)[/tex]
4. [tex]\( D = \left(1+\frac{1}{28}\right)^{28} \approx 2.6712778534408463 \)[/tex]
Next, we need the value of the mathematical constant [tex]\( e \)[/tex], which is approximately [tex]\( e \approx 2.718281828459045 \)[/tex].
Now, let's compare how close each of the evaluated expressions is to [tex]\( e \)[/tex]:
1. Difference for [tex]\( A \)[/tex]:
[tex]\[
|2.6728491439808066 - 2.718281828459045| \approx 0.0454326844782384
\][/tex]
2. Difference for [tex]\( B \)[/tex]:
[tex]\[
|2.6743187758703026 - 2.718281828459045| \approx 0.0439630525887426
\][/tex]
3. Difference for [tex]\( C \)[/tex]:
[tex]\[
|2.6695939778125704 - 2.718281828459045| \approx 0.0486878506464746
\][/tex]
4. Difference for [tex]\( D \)[/tex]:
[tex]\[
|2.6712778534408463 - 2.718281828459045| \approx 0.0470039750181987
\][/tex]
From the differences calculated above, it's evident that the smallest difference is [tex]\( 0.0439630525887426 \)[/tex] which corresponds to the expression [tex]\( B \)[/tex].
Hence, the value of the expression [tex]\( \left(1+\frac{1}{30}\right)^{30} \)[/tex] is closest to [tex]\( e \)[/tex].
