Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = (-1)^{n-1} 5^{n+1}$.

Let the first term of a series be $T_1=6$ and its $r^{\text{th}}$ term $T_r=3T_{r-1}+6^r$ for $r=2, 3, \ldots, n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}(n^2-12n+39)(4 \cdot 6^n - 5 \cdot 3^n + 1)$,then $n$ is equal to:

If the $(m + 1)^{th}$,$(n + 1)^{th}$ and $(r + 1)^{th}$ terms of an $A.P.$ are in $G.P.$ and $m, n, r$ are in $H.P.$,then the value of the ratio of the common difference to the first term of the $A.P.$ is

If $x=\sum_{n=0}^{\infty} \cos ^{2 n} \theta$,$y=\sum_{n=0}^{\infty} \sin ^{2 n} \theta$,$z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \sin ^{2 n} \theta$ and $0 < \theta < \frac{\pi}{2}$,then

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=0, a_{1}=0$ and $a_{n+2}=3a_{n+1}-2a_{n}+1$ for all $n \geq 0$. Then the value of $a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}$ is equal to:

Five numbers are in an $AP$ with a common difference $d \neq 0$. If the $1^{st}$, $3^{rd}$, and $4^{th}$ terms are in a $GP$, then: