Tag: binomial theorem, sequence and series

If a series consists only a finite number of terms it is called a ................

If the sum of first $75$ terms of an AP is $2625$, then the $38^{th}$ term of an AP is

If  in traingle ABC $\cos 2B=\dfrac {\cos (A+C)}{\cos (A-C)}$, then 

A gentlemen invites a party of m + n $(m \neq n)$ friends to a dinner and places m at one table $T _1$ and n at another table $T _2$, the table being round. If not all people shall have the same neighbour n any two arrangement, then the number of ways in which he can arrange the guests, is 

$ \left{ a _ { n } \right} $ and $ \left{ b _ { n } \right} $ are two sequences given by $ a _ { n } = ( x ) ^ { 1 / 2 ^ { \circ } } + ( y ) ^ { 1 / 2 ^ { \circ } } $ and $ b _ { n } = ( x ) ^ { 1 / 2 ^ { 2 } } - ( y ) ^ { 1 / 2 ^ { \circ } } $ for all $ \mathrm { n } \in \mathrm { N } . $ The value of $ \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } \mathrm { a } _ { 3 } \dots \ldots \ldots \mathrm { a } _ { \mathrm { n } } $ is equal to

If $\displaystyle f(n+1)=\frac {2f(n)+1}{2}, n=1,2, .....$ and $f(1)=2$, then $f(101)= ..........$

If $a, b, c$ are in AP, $b - a, c - b$ and $a$ are in GP, then $a : b : c$ is

Let $x _{1}, x _{2}, .....x _{n}$ be in an AP of $x _{1} + x _{4} + x _{9} + x _{11} + x _{20} + x _{22} + x _{27} + x _{30} = 272$, then $x _{1} + x _{2} + x _{3} + ..... + x _{30}$ is equal to

$S _{n} = 1^{3} + 2^{3} + ..... + n^{3}$ and $T _{n} = 1 + 2 + ..... + n$, then

If for $n\in I, n > 10; 1+(1+x)+(1+x)^2+.....+(1+x)^n=\displaystyle\sum^n _{k=0}a _k\cdot x^k, x\neq 0$ then?