Vinod Singh and Shweta Pawar Solutions for Chapter: Sequence and Series, Exercise 3: Competitive Thinking

If $a_1,a_2\dots \dots ,a_n$ are in $\mathrm{AP}$, with common difference, $d$, then, the sum of the following series is $\sin d\left(cosec{a}_1·\mathrm{cosec}{a}_2+cosec{a}_2·cosec{a}_3+\dots \dots +cosec{a}_{n-1}cosec{a}_n\right)$
 

If $S=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{3}\right)-\frac{1}{4}\left(\frac{1}{2^2}+\frac{1}{3^2}\right)+\frac{1}{6}\left(\frac{1}{2^3}+\frac{1}{3^3}\right)$, then $S=$

${99}^{\text{th}}$ term of the series $2+7+14+23+34+\dots .$  is

Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series $1^2+2·2^2+3^2+2·4^2+5^2+2·6^2+\dots \dots$ If $B-2A=100\lambda ,$ then $\mathrm{\lambda }$ is equal to

If $\sin \theta , \cos \theta$ and $\tan \theta$ are in $\mathrm{G}.\mathrm{P}.,$ then ${\cot }^6\theta -{\cot }^2\theta =$

If $\mathrm{tan\theta }=\sqrt{\frac{3}{2}}$, the sum of the infinite series $1+2\left(1-\mathrm{cos\theta }\right)+3{\left(1-\mathrm{cos\theta }\right)}^2+4{\left(1-\mathrm{cos\theta }\right)}^3+\dots .\infty$  is

If $\cos (\mathrm{\theta }-\mathrm{\alpha })$, $\mathrm{cos\theta }$, $\cos (\mathrm{\theta }+\mathrm{\alpha })$ are in harmonic progression, then $\mathrm{cos\theta sec}\left(\frac{\mathrm{\alpha }}{2}\right)$ is equal to

Let $f:N\to N$ be such that $f\left(1\right)=2$ and $f(x+y)=f\left(x\right)f\left(y\right)$ for all natural numbers $x$ and $y$. If $\sum _{k=1}^{n}f(a+k)=16\left(2^n-1\right),$ then $a$ is equal to