Question

Find the common difference of an AP whose first term os 100 and the sum of whose first six term is five times the sum of the next sixterms ?​

Answer 1

Step-by-step explanation:

Let a be the first term and d be a common difference.

Given that the first term is 100.

=> a = 100.

We know that sum of n terms of an AP sn = (n/2)(2a + (n - 1) * d)

Sum of 1st 6 terms:

s6 = (6/2)(2(100) + (6 - 1) * d

     = 3(200 + 5d)

 

     = 600 + 15d.

Sum of next 6 terms:

s12 = (12/2)(2(100) + (12 - 1) * d) - 600 + 15d

      = (6)(200 + 11d) - 600 + 15d

      = 1200 + 66d - 600 + 15d

 

      = 600 + 51d.

Given that sum of 1st 6 terms is 5 times the sum of the next 6 terms.

=> 600 + 15d = 5(600 + 51d)

=> 600 + 15d = 3000 + 255d

=> 600 + 15d - 3000 = 255d 

=> -2400 = 240d

=> d = -10.

Therefore the common difference is d = -10.