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If S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. then P is the ________ of Sn and R-n.
  • a)
    Arithmetic Mean
  • b)
    Geometric Mean
  • c)
    Harmonic Mean
  • d)
    None
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If S be the sum, P the product and R the sum of the reciprocals of n t...
Solution:

Let the first term of the G.P. be a and the common ratio be r.

Then, the n terms are a, ar, ar2, …, arn-1.

Sum of n terms of the G.P. can be calculated using the formula:

Sn = a(1-rn)/(1-r)

Product of n terms of G.P. can be calculated using the formula:

P = a^n rn(n-1)/2

Sum of reciprocals of n terms of G.P. can be calculated using the formula:

R = (1-a^n r-n)/(1-r)

We need to prove that P is the geometric mean of Sn and R-n.

Geometric mean of two numbers a and b is given by the formula:

GM = √ab

Let’s calculate the geometric mean of Sn and R-n.

GM = √(Sn × R-n)

= √[(a(1-rn)/(1-r)) × ((1-r)/(1-a^n r-n))]

= √[a(1-a^n r-n)/(1-a)(1-rn)]

= a^(n/2) rn(n-1)/4

= P^(1/2)

Therefore, we can see that P is the geometric mean of Sn and R-n.

Hence, the correct answer is option (B) Geometric Mean.
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If S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. then P is the ________ of Snand R-n.a)Arithmetic Meanb)Geometric Meanc)Harmonic Meand)NoneCorrect answer is option 'B'. Can you explain this answer?
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