Question

if the sum of the first p terms of an A. P is

$a {p}^{2} + bp$

Find its common difference.

BEST ANSWER WILL BE MARKED AS BRAINLIEST!

Answer 1

Given, Sum of p terms of an AP S(p) = ap^2 + bp.

When p = 1:

⇒ S₁ = a(1)^2 + b(1)

        = a + b.

Hence, S₁ = a₁ = a + b.

When p = 2:

⇒ S₂ = a(2)^2 + b(2)

        = 4a + 2b

Now,

Sum of first two terms = First term + second term .

⇒ S₂ =  a₁ + a₂

⇒ a₂ = S₂ - a₁

    = (4a + 2b) - (a + b)

    = 3a + b

∴ Hence, a₁ = (a + b) and a₂ = (3a + b).

Now,

Common difference = Second term - First term

                                  = 3a + b - a - b

                                  = 2a.

Therefore, Common difference = 2a.

Hope it helps!

Answer 2

$\huge \bf \orange{Hey \: there !! }$


➡ Given :-

→ Sum of first p term of an AP [ $S_p$ ] = ap² + bp .


➡ To find :-

→ The common difference [ d ] .



▶ Solution :-

We have,

$S_p$ = ap² + bp .

Taking p = 1 .

°•° $S_1$ = a(1)² + b(1) .

=> $S_1$ = a + b.

Now, taking p = 2 .

°•° $S_2$ = a(2)² + b(2) .

=> $S_2$ = 4a + 2b.


▶Now, $a_1$ = $S_1$ = a + b .

And, $a_2$ = $S_2$ - $a_1$

= ( 4a + 2b ) - ( a + b ) .

= 4a + 2b - a - b .

= 3a + b .


Then, Common difference (d) = $a_2$ - $a_1$ .

= ( 3a + b ) - ( a + b ) .

= 3a + b - a - b .

$\huge \boxed{ \boxed{ \purple{ \bf = 2a . }}}$ .


✔✔ Hence, the common difference is 2a ✅✅.

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