The sum of the first four terms of an arithmetic progression is 56 and sum of the first eight terms of the same A.P. is 176.find the sum of the first 16 terms of the A.P.

Let the terms be a,a+d,a+2d,a+3d in an AP

Sum of first 4 terms=a+a+d+a+2d+a+3d=4a+6d

Now, Given 4a+6d=56 -------(1)

Next 4 terms including the above 4 terms is 4a+6d+a+4d+a+5d+a+6d+a+7d=176

So, 8a+2d=176 -----------(2)

Solving equation (1) & (2) we obtain the value of d=4

Now, substitute the value of d in equation 1 we get 4a+6.4=56 a=32/4 a=8

So, our A.P series is 8,12,16,20.....

16th term would be a+15d i.e 8+15.4=68

So, our A.P series is 8,12,16,20,.....68

Now since they have asked for the sum of the series hence the formulae is n/2(a+l)

i.e 16/2 (8+68)= 8*76=608 (Ans)