Recent questions tagged sum-of-n-terms-of-special-series

Given $\large\frac{1}{1^4}+\large\frac{1}{2^4}+\large\frac{1}{3^4}$+.....$\infty$ = $\large\frac{{\pi}^{4}}{90}$, then the value of $\large\frac{1}{1^4}+\large\frac{1}{3^4}+\large\frac{1}{5^4}$+....$\infty$ is

asked Jan 23, 2014 by yamini.v

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$a_{1}=50$ and $a_{1}+a_{2}+....+a_{n}=n^2a_{n}$ $\forall\;n \geq 1$, value of $a_{50}$ equal to

asked Jan 23, 2014 by yamini.v

1 answer

Sum of n terms of the series $\large\frac{1}{4}+\large\frac{7}{16}+\large\frac{37}{64}+\large\frac{175}{256}$+... is

asked Jan 23, 2014 by yamini.v

1 answer

Sum of the series $\;\large\frac{2}{5}+\large\frac{3}{5^2}+\large\frac{4}{5^3}+\large\frac{2}{5^4}+\large\frac{3}{5^5}+\large\frac{4}{5^6}+...$ equals:

asked Jan 23, 2014 by yamini.v

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$\sqcap_{n=2}^{\infty}\;\large\frac{n^3-1}{n^3+1}$ equals

asked Jan 23, 2014 by yamini.v

1 answer

If $S_{n}=\large{1}{6}\;n(n+1)(n+2)$ $\forall\;n\;\geq\;1$, then $\displaystyle\lim_{n \to \infty}\;\displaystyle\sum_{r=1}^{n}\;\large\frac{1}{a_{r}}$ equals

asked Jan 23, 2014 by yamini.v

1 answer

Given $1^2+2^2+3^2+....+2003^2$ = $(2003)(4007)(334)$ and $(1)(2003)+(2)(2002)+(3)(2001)+....+(2003)(1)$ = $(2003)(334)(x)$ then value of $x$ is

asked Jan 23, 2014 by yamini.v

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If $a_{n}=\displaystyle\sum_{k=1}^{n}\;\large\frac{1}{k(n+1-k)}$, then for $n\;\geq\;2$

asked Jan 23, 2014 by yamini.v

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If $f_{1}=f_{2}=1$ and thereafter $f_{n+2}=f_{n+1}+f_{n}$ for all $n \in N$. Find $\displaystyle\sum_{n=2}^{\infty}\;\large\frac{1}{f_{n+1}\;.f_{n-1}}$

asked Jan 23, 2014 by yamini.v

1 answer

$S_{1}, S_{2},.....S_{n}$ are sums of infinite geometric series with first term $1,2,3,...n$ and common ratio $\large\frac{1}{2}$, $\large\frac{1}{3}$,...$\large\frac{1}{n+1}$ respectively. Find $\displaystyle\sum_{r=1}^{n}\;S_{r}$.

asked Jan 23, 2014 by yamini.v

1 answer

Find sum upto n terms: $\large\frac{3}{1^2.2^2}+\large\frac{5}{2^2.3^2}+\large\frac{7}{3^2.4^2}$+....

asked Jan 23, 2014 by yamini.v

1 answer

Find the sum of the numbers in the $n^{th}$ set: $(1)$, $(2,3)$, $(4,5,6)$, $(7,8,9,10)$.....

asked Jan 23, 2014 by yamini.v

1 answer

Find sum of first n terms of series : $\;1+\large\frac{1}{1+2}+\large\frac{1}{1+2+3}+\;....$

asked Jan 23, 2014 by yamini.v

1 answer

Evaluate sum to n terms $\;1.1+2.01+3.001+\;.....$

asked Jan 23, 2014 by yamini.v

1 answer

Sum of the series $\;S=\large\frac{4}{7}-\large\frac{5}{7^2}+\large\frac{4}{7^3}-\large\frac{5}{7^4}+.....\;\infty\;$ is :

asked Jan 23, 2014 by yamini.v

1 answer

Given $a_{n}=\displaystyle\sum_{k=1}^{n}\;\sqrt{1+\large\frac{1}{k^2}+\large\frac{1}{(k+1)^2}}\;$ the value of $\;a_{5}\;$ is

asked Jan 23, 2014 by yamini.v

1 answer

Given for every $n \in N \;(1^2-a_{1})+(1^2-a_{2})+.......+(n^2-a_{n})=\large\frac{1}{3}n(n^2-1)\;the\;a_{n}$ equals

asked Jan 23, 2014 by yamini.v

1 answer

Given $\;t_{r}=1^2+2^2+.......\;r^2$ and $t_{1}+t_{2}+t_{3}+...\;t_{n}=\large\frac{k}{12}\;n\;(n+1)\;(n+2)\;$ the value of k will be

asked Jan 23, 2014 by yamini.v

1 answer

$p\;(x)=\large\frac{1+x^2+x^4+\;...\;x^{2n-2}}{1+x+x^2+\;...\;+x^{n-1}}\;$ is a polynomial in x , then n must be

asked Jan 22, 2014 by yamini.v

1 answer

$\alpha$ & $\beta\;$are +ve roots of $\;x^2-2ax+ab=0\;$ then for $\;n \in N\;$$(0 \lt b \lt a)\;S_{n}=1+2(\large\frac{b}{a})$$+3(\large\frac{b}{a})^{\normalsize 2}$$+\;...+\;n$$(\large\frac{b}{a})^{n-1}\;$ can not exceed .

asked Jan 22, 2014 by yamini.v

1 answer

$ If\;a_{1}=\frac{1}{2}\;,a_{k+1}=a_{k}^{2}+a_{k}\;\forall\;k\;\geq\;1\;and\;x_{n}=\large\frac{1}{a_{1}+1}+\large\frac{1}{a_{2}+1}+...\;\large\frac{1}{a_{n}+1}\;the \;value\;of\;[x_{50}]\;is\;([.]\;represents\;greatest\;integer\;function)$)

asked Jan 22, 2014 by yamini.v

1 answer

If $\;a_{1}=1\;,a_{n+1}=2a_{n}+1$ then , $\;a_{n+1}$ equals

asked Jan 22, 2014 by yamini.v

1 answer

Sum of n terms of series $\;S=1^2+2(2)^2+3^2+2(4)^2+5^2+\;....\;$ when n is even is :

asked Jan 22, 2014 by yamini.v

1 answer

$\;A_{1},A_{2}\;,....\;A_{n}\;$ are fixed +ve real number such that $\;A_{1}\;.A_{2}\;..\;A_{n}=k$ , then $\;A_{1}+2A_{2}+\;...\;nA_{n}$ can not be than :

asked Jan 22, 2014 by yamini.v

1 answer

If a,b,c,d are positive real number , then least value of $\;(a+b+c+d)\;(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})\;$ is :

asked Jan 22, 2014 by yamini.v

1 answer

If $p$ = $1$ + $\large\frac{1}{2}$ + $\large\frac{1}{3}$+....+$\large\frac{1}{n}$, then $S$ = $\large\frac{1^2}{1^3}$ + $\frac{1^2+2^2}{1^3+2^3}$ + $\large\frac{1^2+2^2+3^2}{1^3+2^3+3^3}$+.... upto $n$ terms equal to

asked Jan 22, 2014 by yamini.v

1 answer

Sum of n terms of series $(n^2-1^2)$ + $2\;.(n^2-2^2)$ + $3\;.(n^2-3^2)$+.....is

asked Jan 22, 2014 by yamini.v

1 answer

Sum of n terms of series $\;1.3.5+3.5.7+5.7.9+$.......is

asked Jan 22, 2014 by yamini.v

1 answer

Find $\frac{1}{3}+\frac{1}{15}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}$ +....upto n terms

asked Jan 21, 2014 by yamini.v

1 answer

Sum of 20 terms of the series 1+2+3+4+5+8+7+16+9+....... is

asked Jan 21, 2014 by yamini.v

1 answer

The sum $1+\frac{3}{x}+\frac{9}{x^2}+\frac{27}{x^3}+....\infty$, ($x\neq\;0$) is finite if

asked Jan 21, 2014 by yamini.v

1 answer

$\displaystyle\sum_{r=1}^{n}\;\frac{r}{(r+1)!}=$

asked Jan 21, 2014 by yamini.v

1 answer

Sum of $\;\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+\;.....\;upto\;\infty\;is$

asked Jan 21, 2014 by yamini.v

1 answer

$(1-2y)\;(1+3x+9x^2+27x^3+81x^4+243x^5+729x^6)$ = $(1-64y^6)$, $(y\;\neq\;1)$, then $\frac{x}{y}$ is

asked Jan 21, 2014 by yamini.v

1 answer

Sum of $n$ terms of AP is $6n^2+5n$ while $a_{m}=164$, the value of $m$ is

asked Jan 21, 2014 by yamini.v

1 answer

Let $S=\frac{3}{19}+\frac{33}{(19)^2}+\frac{333}{(19)^3}+$......$\infty$. Find $S$.

asked Jan 21, 2014 by yamini.v

1 answer

Natural numbers are divided into groups $\;(1)\;,\;(2,3,4)\;,\;(5,6,7,8,9)\;.....$ Sum of first and last term of $n^{th}\;$ group will be :

asked Jan 21, 2014 by yamini.v

1 answer

Let $a_{n}\;$ be $\;n^{th}$ term of AP. If $\;\displaystyle\sum_{r=1}^{50}\;a_{2r}=p\;and \;\displaystyle\sum_{r=1}^{50}\;a_{2r-1}=q\;$ the common difference is :

asked Jan 21, 2014 by yamini.v

1 answer

If $\;x=\sqrt{2+\sqrt{2+\sqrt{2+------\;\infty}}}\;and\;y=\sqrt{2\;\sqrt{2\;\sqrt{2\;\sqrt{2\;----\;\infty}}}}$ then $xy$ equals

asked Jan 20, 2014 by yamini.v

1 answer

Sum of series $\;S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\;....\;+\frac{1}{100\sqrt{99}+99\sqrt{100}} \;is$

asked Jan 20, 2014 by yamini.v

1 answer

If $1^4+2^4+3^4+.....n^4$ = $an^5+bn^4+cn^3+dn^2+en+f$, find $a$

asked Jan 20, 2014 by yamini.v

1 answer

Sum of series $(n)(n)$ + $(n-1)(n+1)$ + $(n-2)(n+2)$ +....+ $1\;(2n-1)$ is

asked Jan 20, 2014 by yamini.v

1 answer

Sum of series $\;\sum_{r=1}^{n}\;r\;log\;\frac{r+1}{r}$ is :

asked Jan 20, 2014 by yamini.v

1 answer

Sum of $n$ terms of series $S$ = $1$ + $2 \;(1+\frac{1}{n})$ + $3(1+\frac{1}{n})^2$ + ....is given by

asked Jan 20, 2014 by yamini.v

1 answer

For $0\;<\;x\;<\;\pi$, the values of x which satisfies $1+|cos\;x|+|cos\;x|^2+|cos\;x|^3+$....$\infty$ = $2^4$ are

asked Jan 20, 2014 by yamini.v

1 answer

If $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+$......upto $\infty$ = $\frac{\phi^2}{g}$, then $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+$.....upto $\infty$ will be

asked Jan 20, 2014 by yamini.v

1 answer

If $S_{n}=\frac{1}{6}\;n\;(2n^2+9n+13)\;$, then $\sum_{r=1}^n\;\sqrt{a_{r}}\;$ equals.

asked Jan 20, 2014 by yamini.v

1 answer

Sum of series $\;\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....\;is:$

asked Jan 20, 2014 by yamini.v

1 answer

If $f\;(xy)=f\;(x+y)\;\forall\;x,y\;\in\;R$ and $\;f\;(2009)=2009$, then $f\;(-2009)$ equals

asked Jan 20, 2014 by yamini.v

1 answer

Sum of n terms of series $\;\frac{1}{1^3}\;+\frac{1+2}{1^3+2^3}\;+\frac{1+2+3}{1^3+2^3+3^3}\;+....\;is$

asked Jan 20, 2014 by yamini.v

1 answer

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