Q12 of 69 Page 78

If the sum of first p terms of an A.P. is equal to the sum of first q terms then show that the sum of its first (p + q) terms is zero. (p ≠ q)

We know that, sum of nth term of an A.P. we will find it’s

sum



Where, n = no. of terms


a = first term


d = common difference


Sn = sum of n terms


Now, Sum of p terms is



And, Sum of q terms is



Given: Sp = Sq



Multiply by 2 on both sides, we get,


⇒p[ 2a + (p – 1)d] = q[ 2a + (q – 1)d]


⇒2ap + p(p – 1)d = 2aq + q(q – 1)d


⇒ 2ap – 2aq + p(p – 1)d – q(q – 1)d = 0


⇒ 2a(p – q) + d[p2 – p– q2 + q] = 0


⇒ 2a(p – q) + d[(p2– q2 ) – p + q] = 0


⇒ 2a(p – q) + d[(p– q )(p + q) – (p – q)] = 0


(since, (a – b)(a + b) = a2 – b2)


⇒ 2a(p – q) + d(p – q) [p + q – 1 ] = 0


⇒ (p – q)[2a + d (p + q – 1) ] = 0


Since, p ≠ q


∴ p – q ≠ 0


⇒ 2a + d (p + q – 1) = 0


Multiply both side by



⇒ Sp + q = 0


Hence proved


More from this chapter

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There are 37 terms in an A.P., the sum of three terms placed exactly at the middle is 225 and the sum of last three terms is 429. Write the A.P.

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If first term of an A.P. is a, second term is b and last term is c, then show that sum of all terms is

13

If m times the mth term of an A.P. is equal to n times nth term then show that the (m + n)th term of the A.P. is zero.

 14

₹ 1000 is invested at 10 percent simple interest. Check at the end of every year if the total interest amount is in A.P. If this is an A.P. then find interest amount after 20 years. For this complete the following activity.

Simple interest


Simple interest after 1 year


Simple interest after 2 year


Simple interest after 3 year


According to this the simple interest for 4, 5, 6 years will be 400, respectively.


From this and


Amount of simple interest after 20 years


tn + a + (n – 1) d



Amount of simple interest after 20 years is =