Question

If the sum of first p term of an A.P. is ap² + bp, find its common difference.

Answer 1

Answer:

The common difference of p terms of an A.P is 2 a  

Step-by-step explanation:

Given as :

The sum of first p terms of an A.P = a p² + b p

i.e  $S_p$ = a p² + b p

Let The common difference of p terms = d

For first term , i.e $a_1$  =  a (1)² + b × 1

Or,  $a_1$ = a + b              .....1

So, The first term = $a_1$ = a + b

Again

For second term , i.e $a_2$  =  a ( 2 )² + b × 2

Or, $a_2$ = 4 a + 2 b

i.e  $a_1$ + $a_2$  = 4 a + 2 b

Or,   $a_2$ = ( 4 a + 2 b) - $a_1$

Or,   $a_2$ =  ( 4 a + 2 b) - ( a + b )

∴     $a_2$ = 3 a + b        .....2

So, The second term = $a_2$ = 3 a + b

Now, Common difference = d

i.e  common difference = Second term - first term

Or,  d = $a_2$  - $a_1$

Or.  d = ( 3 a + b ) - ( a + b)

∴     d = 2 a

So, The common difference = d = 2 a

Hence, The common difference of p terms of an A.P is 2 a    Answer