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
View Solution FreeMore Application of Derivative Questions
If f(x), for every nu8mber x, then the minimum value of f...If 10 harmonic means H1, H2, H3 .......H10 are inserted between 7 and - , then -...If length of subnormal at any point of the curve yn = an-1x is constant, then n equals-If length of subnormal at any poi...The angle of intersection between the curves y2 = 8x and x2 = 4y − 12 at (2, 4) is -...If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval-...