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Given x, y, z are in G.P. and xp = yq = zσ, then 1/p , 1/q, 1/σ are in
- a)A.P.
- b)G.P.
- c)Both A.P. and G.P.
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Given x, y, z are in G.P. and xp = yq = zσ, then 1/p , 1/q, 1/ ...
Given x, y, z are in G.P. and xp = yq = z, then 1/p , 1/q, 1/ are in A.P.
Explanation:
Let's assume the common ratio of G.P. to be 'r'.
So, we have:
x, xr, xrr, ...
y, yr, yrr, ...
z, z, z, ...
Given, xp = yq = z
=> x^p = y^q = z
=> (x^p)^q = (y^q)^p = z^q
=> x^(pq) = y^(pq) = z^q
Now, we have:
(x^p)^q = x^(pq)
(y^q)^p = y^(pq)
So, x^(pq) = y^(pq) = z^q
=> (x^p)^(1/pq) = (y^q)^(1/pq) = z^(q/pq)
=> x^(1/q) = y^(1/p) = z^(1/pq)
Hence, we can say that 1/p, 1/q, and 1/(pq) are in A.P.
Therefore, option 'A' is the correct answer.
Explanation:
Let's assume the common ratio of G.P. to be 'r'.
So, we have:
x, xr, xrr, ...
y, yr, yrr, ...
z, z, z, ...
Given, xp = yq = z
=> x^p = y^q = z
=> (x^p)^q = (y^q)^p = z^q
=> x^(pq) = y^(pq) = z^q
Now, we have:
(x^p)^q = x^(pq)
(y^q)^p = y^(pq)
So, x^(pq) = y^(pq) = z^q
=> (x^p)^(1/pq) = (y^q)^(1/pq) = z^(q/pq)
=> x^(1/q) = y^(1/p) = z^(1/pq)
Hence, we can say that 1/p, 1/q, and 1/(pq) are in A.P.
Therefore, option 'A' is the correct answer.
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