Let V r denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is ( 2 r - 1 ) . Let T r = V r + 1 - V r - 2 and Q r = T r + 1 - T r for r = 1 , 2 , Q. The sum V 1 + V 2 + ? + V

Question

Let $V_{r}$ denote the sum of first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2 r - 1)$ . Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for $r = 1, 2, \dots$ . [2007]

Q. The sum $V_{1} + V_{2} + \dots + V_{n}$ is

Smart Achievers · Accepted Answer

(2)

$V_{1} + V_{2} + \dots + V_{n} = \sum_{r = 1}^{n} V_{r}$ $= \sum_{r = 1}^{n} (r^{3} - \frac{r^{2}}{2} + \frac{r}{2})$

$= \sum n^{3} - \frac{\sum n^{2}}{2} + \frac{\sum n}{2}$

$= \frac{n^{2} {(n + 1)}^{2}}{4} - \frac{n (n + 1) (2 n + 1)}{12} + \frac{n (n + 1)}{4}$

$= \frac{n (n + 1)}{4} [n (n + 1) - \frac{2 n + 1}{3} + 1]$

$= \frac{n (n + 1) (3 n^{2} + n + 2)}{12}$

Solution

Q.
Let $V_{r}$ denote the sum of first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2 r - 1)$ . Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for $r = 1, 2, \dots$ . [2007]

Q. The sum $V_{1} + V_{2} + \dots + V_{n}$ is

1 $\frac{1}{12} n (n + 1) (3 n^{2} - n + 1)$

2 $\frac{1}{12} n (n + 1) (3 n^{2} + n + 2)$

3 $\frac{1}{2} n (2 n^{2} - n + 1)$

4 $\frac{1}{3} (2 n^{3} - 2 n + 3)$

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Solution

Q. Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let Tr=Vr+1-Vr-2 and Qr=Tr+1-Tr for r=1,2,…. [2007] Q. The sum V1+V2+⋯+Vn is

1 112n(n+1)(3n2-n+1)

2 112n(n+1)(3n2+n+2)

3 12n(2n2-n+1)

4 13(2n3-2n+3)