If-the-first-term-and-n-th-term-of-G-P-are-a-and-b-respectively-p-is-the-product-of-n-terms-Prove-that-p-2-ab-n-

Question Number 47628 by 786786AM last updated on 12/Nov/18

If the first term and n^{th} term of G . P ., are a and b respectively,

p is the product of n terms . Prove that p^{2} = {(ab)}^{n} .

Answered by math1967 last updated on 12/Nov/18

P = a . a r \dots . . {a r}^{n - 1} [r = c o m m o n r a t i o]

P = a^{n} . r^{1 + 2 + \dots . . + (n - 1)}

P = a^{n} . r^{\frac{(n - 1) (n - 1 + 1)}{2}}

P = a^{n} . r^{\frac{n (n - 1)}{2}}

P^{2} = a^{2 n} . r^{n (n - 1)}

P^{2} = {(a . {a r}^{n - 1})}^{n}

∴ P^{2} = {(a b)}^{n} [∵ b = {a r}^{n - 1}]

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