Home
Class 12
MATHS
If a(1),a(2),".....", are in HP and f(k)...

If `a_(1),a_(2),".....",` are in HP and `f_(k)=sum_(r=1)^(n)a_(r)-a_(k)`, then `2^(alpha_(1)),2^(alpha_(2)),2^(alpha_(3))2^(alpha_(4)),"....."` are in `{" where " alpha_(1)=(a_(1))/(f_(1)),alpha_(2)=(a_(2))/(f_(2)),alpha_(3)=(a_(3))/(f_(3)),"....."}`.

A

AP

B

GP

C

HP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
`a_(1),a_(2),a_(3),"......"` are in HP.
`implies (1)/(a_(1)),(1)/(a_(2)),(1)/(a_(3)),"......" " are in AP. " " " "......(i)"`
`:.f_(k)=sum_(r=1)^(n)a_(r)-a_(k)`
`implies a_(k)+f_(k)=sum_(r=1)^(n)a_(r)=lambda " " [" say "]`
`impliesa_(1)+f_(1)=a_(2)+f_(2)=a_(3)+f_(3)="....."=lambda`
From Eq. (i), `(lambda)/(a_(1)),(lambda)/(a_(2)),(lambda)/(a_(3))"......"` are in also in AP.
`(a_(1)+f_(1))/(a_(1)),(a_(2)+f_(2))/(a_(2)),(a_(3)+f_(3))/(a_(3)),"......"` are also in AP.
Subtracting from each term by 1, we get
`(f_(1))/(a_(1)),(f_(2))/(a_(2)),(f_(3))/(a_(3)),"......"` are also in AP.
`(1)/(alpha_(1)),(1)/(alpha_(2)),(1)/(alpha_(3)),"....."` are in AP.
`:.(1)/(alpha_(1)),(1)/(alpha_(2)),(1)/(alpha_(3)),"....."` are in HP.
`2^(alpha_(1)),2^(alpha_(2)),2^(alpha_(3)),"....."` are not in AP/GP/HP.
Promotional Banner

Similar Questions

Explore conceptually related problems

If 1,alpha_(1),alpha_(2),alpha_(3),...,alpha_(n-1) are n, nth roots of unity, then (1-alpha_(1))(1-alpha_(2))(1-alpha_(3))...(1-alpha_(n-1)) equals to

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (a_(3))/(a_(2))+(a_(4))/(a_(3))+"......"+(a_(n))/(a_(n-1))-a_(2)((1)/(a_(2))+(1)/(a_(3))"+........"+(1)/(a_(n-2)))=(a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

Let a_(1),a_(2),"...." be in AP and q_(1),q_(2),"...." be in GP. If a_(1)=q_(1)=2 " and "a_(10)=q_(10)=3 , then

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common rario r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If a_(1),b_(1),c_(1),a_(2),b_(2),c_(2) "and" a_(3),b_(3),c_(3) are three digit even natural numbers and Delta=|{:(c_(1),a_(1),b_(1)),(c_(2),a_(2),b_(2)),(c_(3),a_(3),b_(3)):}| , then Delta is

If 1,alpha_(1),alpha_(2),...,alpha_(n-1) are the n, nth roots of the unity , then find the value of sum_(i=0)^(n-1)(1)/(2-a_(i)).

If a_(m) be the mth term of an AP, show that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+"...."+a_(2n-1)^(2)-a_(2n)^(2)=(n)/((2n-1))(a_(1)^(2)-a_(2n)^(2)) .

If a_1,a_2,a_3….a_r are in G.P then show that |{:(a_(r+1),a_(r+5),a_(r+9)),(a_(r+7),a_(r+11),a_(r+15)),(a_(r+11),a_(r+7),a_(r+21)):}| is independent of r

If a_(i) gt 0 AA I in N such that prod_(i=1)^(n) a_(i) = 1 , then prove that (a + a_(1)) (1 + a_(2)) (1 + a_(3)) .... (1 + a_(n)) ge 2^(n)