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. Which one of the followi

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. Which one of the following is a correct statement

Q. Which one of the following is a correct statement?

Smart Achievers · Accepted Answer

(2)

$Q_{r + 1} - Q_{r} = T_{r + 2} - T_{r + 1} - (T_{r + 1} - T_{r})$

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

$= (r + 3) (3 r + 5) - 2 (r + 2) (3 r + 2) + (r + 1) (3 r - 1)$

$∵$ $Q_{r + 1} - Q_{r} = 6 (r + 1) + 5 - 6 r - 5 = 6 (constant)$

$∴$ $Q_{1}, Q_{2}, Q_{3}, \dots are in A.P. with common difference 6 .$

Solution

1 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 5

2 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 6

3 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 11

4 $Q_{1} = Q_{2} = Q_{3} = \dots$

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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. Which one of the following is a correct statement

1 Q1,Q2,Q3,… are in A.P. with common difference 5

2 Q1,Q2,Q3,… are in A.P. with common difference 6

3 Q1,Q2,Q3,… are in A.P. with common difference 11

4 Q1=Q2=Q3=⋯

Ans. (2) Qr+1-Qr=Tr+2-Tr+1-(Tr+1-Tr) =Tr+2-2Tr+1+Tr =(r+3)(3r+5)-2(r+2)(3r+2)+(r+1)(3r-1) ∵ Qr+1-Qr=6(r+1)+5-6r-5=6 (constant) ∴ Q1,Q2,Q3,… are in A.P. with common difference 6.

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1 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 5

2 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 6

3 $Q_{1}, Q_{2}, Q_{3}, \dots$ are in A.P. with common difference 11

4 $Q_{1} = Q_{2} = Q_{3} = \dots$