Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Manipal MET2012MathematicsSequences and Series

If x 1, y 1, z 1 are in GP, then 1 1+ x 1 1+ y , 1 1+ z are in

Options

  1. AAP
  2. BHP
  3. CGP
  4. DNone of these

Correct answer

B. HP

Step-by-step solution

Let the common ratio of the GP be r . Then, y=x r and z=x r^2 . y= x+ r and z= x+2 r Putting A=1+ x, D= r Then, 1 1+ x = 1 A , aligned & 1 1+ y = 1 1+ x r = 1 1+ x+ r = 1 A+D & and 1 1+ z = 1 1+ x+2 r = 1 A+2 D aligned Here we see that A, A+D and A+2 D are in AP. 1 A , 1 A+D and 1 A+2 D are in HP. Therefore, 1 1+ x , 1 1+ y , 1 1+ z , are in HP.

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Every term of a geometric progression is positive , and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is: 2026If k is the arithmetic mean of two given quantities and p, q are the geometric means between the same two quantities, then p^3 + q^3 is: 2026Let ' a ' and ' b ' be two numbers where a < b . The geometric mean of these numbers exceeds the smaller number by 12 and the arithmetic mean is smaller than the larger number by 2 2026The product of three numbers in geometric progression is 8 and the sum of the product of the numbers taken in pairs is 14. Find the numbers. 2026Let a₁, a₂, a₃, be a geometric progression of positive integers such that a₁ = 3 and a_ n+2 - 2a_n = a_ n+1 for all positive integers n . What is the value of a₁ + a₂ + a₃ + a₄ + a 2026The angles of a triangle are in A.P and the greatest angle is double the least angle, then sine of the third angle is 2026If we insert two numbers between 2 and 4 so that the resulting sequence is in G.P, then the inserted numbers in the order are 2026If S _ n =1^3+2^3+ . .+ n ^3 and T _ n =1+2+ . .+ n , then 2025 Full Sequences and Series list All Manipal MET PYQs