IIMB 2002 (QA) PYQs

Solve:

$\dfrac{1}{1\times2\times3\times4}+\dfrac{1}{2\times3\times4\times5}+\dfrac{1}{3\times4\times5\times6}+\cdots+\dfrac{1}{15\times16\times17\times18}$

If $\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$ is the arithmetic mean of $a$ and $b$ then find the value of $n$

The sum of first $9$ terms of the series $\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots .$ is