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Question

# How many terms of the A.P. are needed to give the sum –25?

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Solution

## The given A.P. is −6,− 11 2 ,−5,…. Let a, d be the first term and common difference of the given A.P. a=−6 d=− 11 2 +6 = 12−11 2 = 1 2 Let the sum of n terms of the given A.P. be −25. The formula for the sum of n terms in an A.P. is given by, S n = n 2 [ 2a+( n−1 )d ] Substitute the values of a, dand S n as −6, 1 2 , −25 in the above expression. −25= n 2 [ 2×( −6 )+( n−1 )×( 1 2 ) ] −25×2=n[ −12+ n 2 − 1 2 ] −50=n[ − 25 2 + n 2 ] −50×2=n( −25+n ) Further simplify the above expression. −100=n( −25+n ) 100=n( n−25 ) 100= n 2 −25n n 2 −25n−100=0 Further simplify the above equation. n 2 −25n−100=0 n 2 −5n−20n−100=0 n( n−5 )−20( n−5 )=0 ( n−20 )( n−5 )=0 Equate the above expression to obtain the value of n. n=20 or 5. Thus, the total number of terms in the A.P. −6,− 11 2 ,−5,…is 20 or 5.

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