Let's evaluate each expression and compare their values to the mathematical constant [tex]\( e \)[/tex], which is approximately [tex]\( 2.718281828459045 \)[/tex].
1. Expression A: [tex]\(\left(1+\frac{1}{12}\right)^{12}\)[/tex]
We calculate:
[tex]\[
\left(1+\frac{1}{12}\right)^{12} \approx 2.613035290224676
\][/tex]
2. Expression B: [tex]\(\left(1+\frac{1}{14}\right)^{14}\)[/tex]
We calculate:
[tex]\[
\left(1+\frac{1}{14}\right)^{14} \approx 2.6271515563008685
\][/tex]
3. Expression C: [tex]\(\left(1+\frac{1}{13}\right)^{13}\)[/tex]
We calculate:
[tex]\[
\left(1+\frac{1}{13}\right)^{13} \approx 2.6206008878857308
\][/tex]
4. Expression D: [tex]\(\left(1+\frac{1}{11}\right)^{11}\)[/tex]
We calculate:
[tex]\[
\left(1+\frac{1}{11}\right)^{11} \approx 2.6041990118975287
\][/tex]
Next, we find the difference between each calculated value and [tex]\( e \)[/tex]:
1. Difference for Expression A:
[tex]\[
|2.613035290224676 - 2.718281828459045| \approx 0.105246538234369
\][/tex]
2. Difference for Expression B:
[tex]\[
|2.6271515563008685 - 2.718281828459045| \approx 0.0911302721581765
\][/tex]
3. Difference for Expression C:
[tex]\[
|2.6206008878857308 - 2.718281828459045| \approx 0.0976809405733142
\][/tex]
4. Difference for Expression D:
[tex]\[
|2.6041990118975287 - 2.718281828459045| \approx 0.1140828165615163
\][/tex]
Upon comparing these differences:
- The difference for Expression A is approximately [tex]\( 0.1052 \)[/tex]
- The difference for Expression B is approximately [tex]\( 0.0911 \)[/tex]
- The difference for Expression C is approximately [tex]\( 0.0977 \)[/tex]
- The difference for Expression D is approximately [tex]\( 0.1141 \)[/tex]
We see that the smallest difference from [tex]\( e \)[/tex] is for Expression B.
Therefore, the value of the expression [tex]\(\left(1+\frac{1}{14}\right)^{14}\)[/tex] is closest to [tex]\( e \)[/tex].
