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The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is formed by multiplying each of the first 2n terms by the next term. Find the sum of new series
Asked by rushabhjain.av | 02 Nov, 2019, 10:10: PM
First term = a and last term = b
Let d be the common difference of an A.P.
The new series formed is
a(a+d), (a+d)(a+2d), (a+2d)(a+3d), ....., (a+2n)b, b
Sum of this new series
= a(a+d) + (a+d)(a+2d) + (a+2d)(a+3d) +.....+ (a+2n)b + b
= (a+d)(a+a+2d) + (a+3d)(a+2d+a+4d) + ... + (a+(2n-1)d)(a+(2n-2)d+a+2nd)+b
= 2(a+d)2 + 2(a+3d)2 + .... + 2(a+(2n-1)d)2 + b
= 2[(a+d)2 + (a+3d)2 + .... + (a+(2n-1)d)2]+b
a+d, a+3d, ... a+(2n-1)d forms an A.P. with first term a+d and common difference 2d
Now, use the formula for sum of squares of n terms of an A.P.
, where a+d is the first term, 2d is the common difference and n is the number of terms
Answered by Renu Varma | 07 Nov, 2019, 11:02: AM

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