Sequences and Series

Let S be the sum of the first 9 terms of series (x+ka)+(x²+(k+2)a)+(x³+(k+4)a)+(x?+(k+6)a)+., where a≠0 and x≠1. If S=(x¹?-x+45a(x-1))/(x-1), then k is equal to

If the sum of first 11 terms of an A.P. a?, a?, a?, . is 0 (a? ≠ 0), then the sum of the A.P. a?, a?, a?, ., a? is ka?, where k is equal to

The sum of the first three terms of a G.P is S and their product is 27. Then all such S lie in

Let α and β be the roots of the equation, 5x²+6x-2=0. If Sₙ=αⁿ+βⁿ, n=1,2,3,., then

The value of λ and µ for which the system of linear equations
x+y+z=2
x+2y+3z=5
x+3y+λz=µ
has infinitely many solutions are, respectively:

If α, β are roots of the equation x² + 5√2x + 10 = 0, α > β and P? = α? - β? for each positive integer n, then the value of (P?P? + 5√2P?P?)/(P?P? + 5√2P?²) is equal to……….

Let S? be the sum of the first n terms of an arithmetic progression. If S? = 3S?, then the value of S?/S? is:

The ${}_{}^{}$ $\mathrm{s}\mathrm{u}\mathrm{m}?{\sum }_{\mathrm{k}=1}^{20}?(1+2+3+\dots .\mathrm{k})$ is(Series)

Let $f:R\to R$ $$ be such that for all $x\in R\left(2^{1+x}+2^{1-x}\right),f\left(x\right)$ $\left({}^{}{}^{}\right)$ and $\left(3^x+3^{-x}\right)$ $\left({}^{}{}^{}\right)$ are in A.P., then the minimum value of $f\left(x\right)$ $$ is: