how many terms of AP:20,58/3,56/3,....should be taken so that their sum is 300.Explain the double answer.

Asked by satyendra | 20th Feb, 2014, 01:02: PM

Expert Answer:

Dear Student,

 AP : 20, 58 / 3 , 56 / 3..............

First term (a) = 20

Common Difference (d) = (58 / 3) - 20 =  (-2 / 3)

Sum of n term (S n) = 300

Apply : Sum of n term (S n)  = [n/2] x(2a + (n-1)xd )

300 = [n/2] x 2x{20} + (n-1)x{-2 / 3}  )

300 * 2 = n ( 40 + [- 2 / 3] n - [-2 / 3] )

600 = n ( 40 x 3 - 2 n +2)  /  3

1800 = n ( 120 + 2 - 2n )

1800 = n ( 122 - 2n)

Divide the whole equation by "2"

900 = n ( 61 - n )

n2 - 61n + 900 = 0

n2 +(- 61n) + 900 = 0

Apply Splitting the middle term : 900 = -36 x -25

n2 +(-36n - 25n+ 900 = 0

n ( n - 36) -25 (n - 36) = 0

So : (n-36) x ( n - 25) = 0

=> ( n -36 ) = 0 OR ( n - 25) = 0

=>The total Number of term is (n) = 36  or 25

The AP IS IN A DESCENDING ORDER SO THE DOUBLE ANS CAN BE EXPLAINED IN TERMS OF THE SUM OF N TERMS.

SINCE THE AP IS IN A DESCENDING THERE IS A POSSIBILITY FOR NEGETIVE NUMBERS TO OCCUR IN THE AP SO THAT IS WHY THE SUM OF N TERM WHICH IS 300 CAN  BE OF FOR FIRST 26 TERMS OR FOR FIRST  35 TERMS OF AP .

That is why there is a occourence of double.

Thanks and Regards

Toppers Learning

Answered by  | 20th Feb, 2014, 01:40: PM

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