Question

If the sum of the first p terms of an ap is a p square + b p find its common difference

Answer 1

The common difference is 2a.

• Sum of AP formula is given by $S_n$ = $\frac{n}{2}(2a + (n-1)d)$ , where a' is the first term , d is the common difference and n is the number of terms.

• It is given that sum of p terms of AP is equal to $ap^{2} + bp$ .

• Sum of n terms of AP ( $S_n$ ) = $\frac{n}{2}(2a + (n-1)d)$ .

According to the given condition ,

$\frac{p}{2}(2a' + (p-1)d)$ =  $ap^{2} + bp$

$a'p + \frac{p^{2}}{2} - \frac{pd}{2}$ = $ap^{2} + bp$

$a' + \frac{pd}{2} -d = ap + b$

• The above equation is a equation in one variable 'p'.
• Comparing both sides we get,

a' - d =b and $\frac{d}{2}$ = a

• We have to find the common difference(d) ,

$\frac{d}{2}$ = a

d = 2a