The sum of the infinite series $$1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \over {{6^6}}} + \,\,.....$$ is equal to :

Let $$\{ {a_n}\} _{n = 0}^\infty $$ be a sequence such that $${a_0} = {a_1} = 0$$ and $${a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1$$ for all n $$\ge$$ 0. Then, $$\sum\limits_{n = 2}^\infty {{{{a_n}} \over {{7^n}}}} $$ is equal to:

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

Let A₁, A₂, A₃, ....... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = $${1 \over {1296}}$$ and A₂ + A₄ = $${7 \over {36}}$$, then the value of A₆ + A₈ + A₁₀ is equal to