Sequences and Series

If $S_n=1^3+2^3+\ldots+n^3$ and $T_n=1+2+\ldots+n$, then

$\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots$ to 24 terms $=$

If $t_n=\frac{1}{4}(n+2)(n+3), n \in N$, then which one of the following is true?
Assertion (A) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_{2003}}=\frac{2003}{3009}$
Reason (R) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_n}=\frac{4 n...

If $x>\sqrt{3}$ and $\frac{x^2+1}{\left(x^2+2\right)\left(x^2+3\right)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is

The sum of all integers between 1 and 100 (both inclusive) which are divisible by 5 or 13 is

\(1+\frac{4}{15}+\frac{4 \cdot 10}{15 \cdot 30}+\frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45}+\ldots . .+\infty=\)

For all $n \in N, \frac{3^n-1}{2} \geq$

\(\sum\limits_{k=1}^n k(k+1)(k+2) \ldots(k+r-1)=\)

If $2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots$ to $n$ terms $=a n^3+b n^2+c n+d$, then $a-b+c-d=$

For all $n \in N$, if $1^3+2^3+3^3+\ldots n^3>x$, then a value of $x$ among the following is

The $n$th term of the series $1+(3+5+7)+(9+11+13+15+17)+\ldots$ is

The number of ways of selecting- 3 numbers that are in GP from the set $\{1,2,3$, $100\}$ is

$|x|<1$, The coefficient of $x^2$ in the power series expansion of $\frac{x^4}{(x+1)(x-2)}$ is

If $\alpha, \beta$ are the roots of the equation $x^2-6 x-2=0$, $\alpha>\beta$ and $a_n=\alpha^n-\beta^n, n \geq 1$, then the value of $\frac{a_{10}-2 a_8}{2 a_9}$ is equal to

\(2+3+5+6+8+9+\ldots .2 n \text { terms }=\)

If the roots of the equation $4 x^3-12 x^2+11 x+m=0$ are in arithmetic progression, then $m=$

$\frac{1}{1 \cdot 5}+\frac{1}{5 \cdot 9}+\frac{1}{9 \cdot 13}+\ldots$. upto $n$ terms $=$

If $1 \cdot 3 \cdot 5+3 \cdot 5 \cdot 7+5 \cdot 7 \cdot 9+\ldots n$ terms $=n(n+1) f(n)-3 n$, then $f(l)=$

The condition that the roots of $x^3-b x^2+c x-d=0$ are in arithmetic progression is

In the expansion of $\frac{2 x+1}{(1+x)(1-2 x)}$ the sum of the coefficients of the first 5 odd powers of $x$ is

\(1+\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9}+\ldots \text { to } \infty=\)

$ \frac{1}{3 \cdot 7}+\frac{1}{7 \cdot 11}+\frac{1}{11 \cdot 15}+\ldots$ to 50 terms $=$

If the roots of the equation $x^3+a x^2+b x+c=0$ are in arithmetic progression. Then,

If $2 \cdot 4^{2 n+1}+3^{3 n+1}$ is divisible by $k$ for all $n \in N$, then $k=$

\(2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots \text { to } 10 \text { terms }=\)

\(1-\frac{2}{3}+\frac{2 \cdot 4}{3 \cdot 6}-\frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9}+\ldots \infty=\)

If the roots of equation $x^3-13 x^2+K x-27=0$ are in geometric progression, then $K=$

Suppose that the three points \(A, B\) and \(C\) in the plane are such that their \(x\)-coordinates as well as \(y\)-coordinates are in GP with the same common ratio. Then, the points \(A, B\) and \(C\)

Using mathematical induction, the numbers \(a_n^{\prime}\) s are defined by \(a_0=1, a_{n+1}=3 n^2+n+a_n (n \geq 0)\), then \(a_n\) is equal to

Let \(f(x)=x^3+a x^2+b x+c\) be polynomial with integer coefficients. If the roots of \(f(x)\) are integer and are in Arithmetic Progression, then \(a\) cannot take the value

Let \(p\) and \(q\) be the roots of the equation \(x^2-2 x+A=0\) and let \(r\) and \(s\) be the roots of the equation \(x^2-18 x+B=0\). If \(p < q < r < s\) are in AP then the values of \(A\) and \(B\) are

If \(1+x^2=\sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2\) is equal to