If $a = r + r^2 + r^3 + \dots + \infty$,then the value of $r$ is .......

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2) = 0, n \in R$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the $G$.$P$.,whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is:

If three numbers are in a geometric progression,then their logarithms are in:

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

What will $Rs. 500$ amount to in $10$ years after its deposit in a bank which pays an annual interest rate of $10\%$ compounded annually?

Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1)=3$ and $\sum_{k=1}^{n} f(k)=3279$,then the value of $n$ is: