Question

find the common difference of a.p whose sum of n terms is given sn = -2n2+n

Answer 1

GIVEN :

The Arithmetic progression whose sum of n terms is given $S_n = -2n^2+n$

TO FIND :

The common difference of A.P

SOLUTION :

Given the Arithmetic progression whose sum of n terms is given $S_n = -2n^2+n$

$S_n = -2n^2+n$

We now that $a_n = S_n - S_{n-1}$

Put n=n-1 in $S_n = -2n^2+n$ we get

$S_{n-1} = -2(n-1)^2 + (n-1)$

By using the Algebraic formula

$(a-b)^2=a^2-2ab+b^2$

$= -2(n^2-2n+1)+n-1$

$=-2n^2+4n-2+n-1$

$S_{n-1}=-2n^2+5n-3$

Substitute the values in $a_n = S_n - S_{n-1}$

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

$a_n=-2n^2+n-(-2n^2+5n-3)$

$=-2n^2+n+2n^2-5n+3$

$=-4n+3$

$a_n=-4n+3$

Put n = 1 in above equation we get,

$a_1=-4(1)+3$

$=-4+3$

$a_1=-1$

Put n = 2 in $a_n=-4n+3$

$a_2=-4(2)+3$

$=-8+3$

$a_2=-5$

Put n = 3 in $a_n=-4n+3$

$a_3=-4(3)+3$

$=-12+3$

$a_3=-9$

The common difference $d=a_2-a_1$

$d=-5-(-1)$

$=-5+1$

d=-4

$d=a_3-a_2$

$=-9-(-5)$

$=-9+5$

d=-4

∴ common difference d is -4

Answer 2

Answer:

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