Question

How many terms of the A.P 45,39,33,... must be taken so that their sum is 180? Explain the double answer.

Answer 1

Answer:

6  & 10

Step-by-step explanation:

How many terms of the A.P 45,39,33,... must be taken so that their sum is 180? Explain the double answer

First Term = a = 45

common difference d = 39 -45 = -6

Sum of n terms = (n/2)(a + a + (n-1)d)

=(n/2) (45 + 45 + (n-1)(-6))

= (n/2)(90 -6n + 6)

=(n/2)(96 - 6n)

= n(48 - 3n)

= 3n(16-n)

Sum given = 180

=> 3n(16-n) = 180

=> n(16-n) = 60

=> 16n - n² = 60

=> n² - 16n + 60 = 0

=> n² - 6n - 10n - 60 = 0

=> n(n-6) -10(n-6) = 0

=> (n-10)(n-6) = 0

=> n = 6  & n = 10

6  or 10 Terms must be taken to make sum = 180

Reason for  the double answer

First six terms are positive and make sum = 180

then 7th & 8th terms are positive & 9th & 10th terms are negative which cancels each other and make sum again 180

Answer 2

Answer:

Sum = 6 or 10

Step-by-step explanation:

Sn = 180 = n /2

[ ] 90 + − ( ) 1 6 ( ) − 1

Þ 360 = 90n – 6n

2

+ 6n