Question

If the sum of first n terms of an AP is An + Bn² where A and B are constants, the common difference of AP will be

Answer 1

Answer:

If the sum of first n terms of an A.P. is pn + qn2 where p and q are constants, then common difference of A.P. will be. a). p + q. b). p - q. c). 2 p. d). 2 q. Correct answer is option 'D'.

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

If in an arithmetic progression

1.

$t_n = n \: th \: term$

$S_n = Sum \: of \: the \: first \: n \: terms$

Then

$t_n = S_n - S_{n-1}$

2.

Common Difference = Second Term - First Term

GIVEN

If the sum of first n terms of an AP is An + Bn²

where A and B are constants

TO DETERMINE

The common difference of the Arithmetic progression

EVALUATION

Let

$t_n = n \: th \: term$

$S_n = Sum \: of \: the \: first \: n \: terms$

$S_n =An + B {n}^{2}$

Then

$t_n = S_n - S_{n-1}$

$\implies \: t_1 = S_1 = A + B$

Also

$S_2 = 2A + B \times {2}^{2} = 2A +4 B$

So

$t_2 = S_2 - S_{1}$

$\implies \: t_2= A + \: 3 B$

So

Common Difference

$= t_2 - t_1$

$=( A + \: 3 B \: ) - (A + \: B)$

$= 2 B$