CBSE Class 12-science Answered
If the sum of first three terms of an ap is a and sum of last three terms is b. If n be the no. Of terms then show that the sum of all terms is given by Sn=n(a+b)/6
Asked by rushabhjain.avv | 18 Nov, 2018, 04:49: PM
Expert Answer
If three numbers are in A.P , then middle number is one-third of sum of these three numbers.
This we can verify, if we assume the three numbers are a-d, a and a+d so that middle number is one-third of sum 3a.
Let A be the first number in A.P of n numbers and the common difference is d.
We are given that sum of first three numbers in A.P is a.
Hence 2nd term of A.P is given by, A+d = a/3 ...................(1)
Similarly if sum last three numbers in A.P is b,
(n-1)th term is given by, A+(n-2)d = b/3 ..................(2)
By subtracting eqn.(1) from eqn.(2), and solving for d, we get
...........................(3)Now sum S of A.P is given by , S = (n/2) [ 2×A + (n-1) d ] = (n/2) [ 2 (A+d) + (n-1)d - 2d ] = (n/2)[ 2(A+d) +(n-3)d ] .................(4)
substituting in eqn.(4) for (A+d) from eqn.(1) and for (n-3)d from eqn.(3), we get, S = (n/2) [ 2a/3 +(b-a)/3 ] = n(a+b)/6
Answered by Thiyagarajan K | 19 Nov, 2018, 12:09: AM
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