If in an A.P., Sn = n²p and Sm = m²p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
A. (1/2)p³
B. mn p
C. p³
D. (m + n) p²
Answers
answer : option (C) p³
given, Sn = n²p and Sm = m²p
here Sr denotes the sum of r terms of the ap.
we have to find Sp
Let a is the first term and d is the common difference of the ap.
Sn = n²p
⇒n/2 [2a + (n -1)d ] = n²p
⇒2a + (n - 1)d = 2np ........(1)
Sm = m²p
⇒m/2 [2a + (m - 1)d ] = m²p
⇒2a + (m - 1)d = 2mp .........(ii)
from equations (1) and (2) we get,
[2a + (n - 1)d]/[2a + (m - 1)d ] = 2np/2mp
⇒[2a + (n - 1)d ] × m = [2a + (m - 1)d ] × n
⇒2am + (mn - m)d = 2an + (mn - n)d
⇒2a(m - n) + (mn - m - mn + n)d = 0
⇒2a(m - n) = (m - n)d
⇒2a = d ...,..........(3)
from equations (1) and (3),
d = 2p = 2a
⇒a = p and d = 2p
now Sp = p/2[2a + (p - 1)d ]
= p/2 [ 2p + (p - 1)2p ]
⇒p/2 × 2p²
= p³
hence option (C) is correct choice.
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