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. T r is always [

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. $T_{r}$ is always

Smart Achievers · Accepted Answer

(4)

$T_{r} = V_{r + 1} - V_{r} - 2$

$= [{(r + 1)}^{3} - \frac{{(r + 1)}^{2}}{2} + \frac{r + 1}{2}] - [r^{3} - \frac{r^{2}}{2} + \frac{r}{2}] - 2$

$= 3 r^{2} + 2 r + 1 = (r + 1) (3 r - 1)$

For each $r, T_{r}$ has two different factors other than 1 and itself.

$∴$ $T_{r}$ is always a composite number.

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. $T_{r}$ is always

1 an odd number

2 an even number

3 a prime number

4 a composite number

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