Question

How many terms of the AP 16, 14, 12,... are needed to give the sum 60? Explain why do we get two answers. on the same​

Answer 1

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Answer 2

Step-by-step explanation:

Here S n = 60 , a = 16 , d = − 2 , n = ? Sn=60,a=16,d=-2,n=? S h n = n 2 [ ( 2 a + ( n − 1 ) d ] Shn=n2[(2a+(n-1)d]………..(Formula) ∴ 60 = n 2 [ 2 × 16 + ( n − 1 ) × ( − 2 ) ] ∴60=n2[2×16+(n-1)×(-2)]….[Substitutig the values] ∴ 60 = n 2 ( 32 − 2 n + 2 ) ∴60=n2(32-2n+2) ∴ 60 = n 2 ( 34 − 2 n ) ∴60=n2(34-2n) ∴ 60 = n 2 × 2 ( 17 − n ) ∴60=n2×2(17-n) ∴ 60 = n ( 17 − n ) ∴60=n(17-n) ∴ 60 = 17 n − n 2 ∴60=17n-n2 ∴ n 2 − 17 n + 60 = 0 ∴n2-17n+60=0 ∴ n 2 − 5 n − 12 n + 60 = 0 ∴n2-5n-12n+60=0 ∴ n ( n − 5 ) − 12 ( n − 5 ) = 0 ∴n(n-5)-12(n-5)=0 ∴ ( n − 5 ) ( n − 12 ) = 0 ∴(n-5)(n-12)=0 ∴ n − 5 = 0 ∴n-5=0 or n − 12 = 0 n-12=0 ∴ n = 5 ∴n=5 or n = 12 n=12 The required terms are 5 or 12. Explanation: THe common difference d of the A.P. is − 2 -2 ∴ ∴ The terms of the A.P. are in descending order. Taking n = 5 n=5 the first 5 terms are 16,14,12,10,8. The sum is 60. Taking n = 12 n=12, the last 7 terms ( 12 − 5 ) (12-5) are 6 , 4 , 2 , 0 , − 2 , − 4 , − 5 6,4,2,0,-2,-4,-5 The sum of these seven terms is 0. ∴ ∴ The sum of first 12 terms is also 60. The sum of the first terms = = the sum of the first twelve terms. ∴ ∴ we get two answers. Ans. 5 terms or 12 terms.Read more on Sarthaks.com - https://www.sarthaks.com/1215255/how-many-terms-of-the-p-16-14-12-are-needed-to-given-the-sum-60-explain-why-do-we-get-two-answers