Application of DerivativeHard
Question
Let a1, a2, a3 ...... and b1, b2, b3 ........ be arithmetic progression such that a1 = 25, b1 = 75 and a100 + b100 = 100, then -
Options
A.The common difference in progression ′ai′ is equal but opposite in sign to the common difference in progression ′bj′.
B.an + bn = 100 for any n.
C.(a1 + b1), (a2 + b2), (a3 + b3) , ...... are in A.P.
D.
(ar + br) = 104
(ar + br) = 104Solution
Let a1, a1 + d1, a1 + 2d1 ...... and b1, b1 + d2, b1 + 2d2, ......be two A.P.′s
∴ a100 = a1 + 99d1, b100 = b1 + 99 d2
Adding a100 + b100 = a1 + b1 + 99 (d1 + d2)
or a100 = 100 + 99(d1 + d2)
⇒ d1 + d2 = 0 or d1 = - d2
∴ option (B) gives an + bn
= a1 + (n -1) d1 + b1 + (n - 1)d2
= a1 + b1 = 100
option (C) is obviously true.
Now
(ar + br) = 100 (a1 + b1) = 104
∴ a100 = a1 + 99d1, b100 = b1 + 99 d2
Adding a100 + b100 = a1 + b1 + 99 (d1 + d2)
or a100 = 100 + 99(d1 + d2)
⇒ d1 + d2 = 0 or d1 = - d2
∴ option (B) gives an + bn
= a1 + (n -1) d1 + b1 + (n - 1)d2
= a1 + b1 = 100
option (C) is obviously true.
Now
(ar + br) = 100 (a1 + b1) = 104Create a free account to view solution
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