ISI2015-MMA-65

Question

Let $n$ be a positive real number and $p$ be a positive integer. Which of the following inequalities is true?

• A. $n^p > \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$
• B. $n^p < \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$
• C. $(n+1)^p <  \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$
• D. none of the above

Answer 1

$$\begin{align}(n+1)^{p+1} - n^{p+1} &= \left(n^{p+1} + \dbinom{p+1}{1}n^p + \cdots + 1\right) - n^{p+1}\\
&=  \dbinom{p+1}{1}n^p + \cdots + 1 \\
&> \dbinom{p+1}{1}n^p & (\because n > 0) \\
&= (p+1)n^p.\end{align}$$

Thus,

$$n^p < \dfrac{(n+1)^{p+1} - n^{p+1}}{p+1}.$$

So option (b) is correct