Let V$$_r$$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ($$2r-1$$). Let $${T_r} = {V_{r + 1}} - {V_r} - 2$$ and $${Q_r} = {T_{r + 1}} - {T_r}$$ for r = 1, 2, ...

T$$_r$$ is always

Let V$$_r$$ denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ($$2r-1$$). Let $${T_r} = {V_{r + 1}} - {V_r} - 2$$ and $${Q_r} = {T_{r + 1}} - {T_r}$$ for r = 1, 2, ...

Which one of the following is a correct statement?

If total number of runs scored in $$n$$ matches is $$\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$$ where $$n > 1$$, and the runs scored in the $$k^{\text {th }}$$ match are given by $$k .2^{n+1-k}$$, where $$1 \leq k \leq n$$. Find, $$n$$.