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# If the Sum of First P Term of an A.P. is Ap2 + Bp, Find Its Common Difference. - Mathematics

Sum

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

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#### Solution

Here, we are given,

Sp = ap2 + bp

Let us take the first term as a’ and the common difference as d.

Now, as we know,

ap = Sp - Sp -1

So, we get,

a_p = (ap^2 +bp) - [a(p - 1)^2 + b (p-1)]

= ap^2 + bp - [a(p^2 + 1 -2 p) + bp - b]        [\text{Using} (a - b)^2 = a^2 + b^2 - ab]

 = ap^2 + bp - (ap^2 + a - 2ap + bp -b)

=ap^2 + bp - ap^2 - a + 2ap - bp +b

=2ap - a + b                                ..............(1)

Also,

a_p = a' + (p-1)d

= a' + pd - d

= pd + ( a' - d)                        ..............(2)

On comparing the terms containing in (1) and (2), we get,

dp = 2ap

d = 2a

Therefore, the common difference is   d = 2a .

Is there an error in this question or solution?
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Q 15 | Page 56
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