Question

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289 then find the common difference of the AP.

Answer 1

Let, 'a' be the first term and 'd' be the common difference of the AP.
$\because$ Sum of n terms of an AP is:
$\frac{n}{2}\times$(2a + (n-1)d)
$\therefore$ Sum of first 7 terms
= $\frac{7}{2}\times$(2a + (7-1)d)
= $\frac{7}{2}\times$ (2a + 6d)
= 7 $\times$ (a+3d)
and Sum of first 17 terms
= $\frac{17}{2}\times$(2a + (17-1)d)
= $\frac{17}{2}\times$ (2a + 16d)
= 17 $\times$ (a + 8d)
Given, Sum of first 7 terms = 49 and Sum of first 17 terms = 289
$\therefore$ 7 $\times$ (a+3d) = 49
$\Rightarrow$ a + 3d = 7 ......(1)
and, 17 $\times$ (a + 8d) = 289
$\Rightarrow$ a + 8d = 17.........(2)
Subtracting (1) from (2), we get
5d = 10
$\Rightarrow$ d = 2
$\therefore$ Common difference of the given AP is 2.