If the first and the nth term of a G.P. are ...

NCERT Class XI Mathematics - Sequences and Series - Solutions

Question : 57 of 106

Marks: +1, -0

If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that

P^2

(ab)^n

Solution:

Let r be the common ratio of the given G.P., then b = nth term =

ar^{n-1}

⇒

r^{n-1}

\frac{b}{a}

⇒ r =

\left(\frac{b}{a}\right)^{\frac{1}{n-1}}

Now, P = product of the first n terms

⇒ P =

a \cdot ar \cdot ar^{2}

...

ar^{n-1}

a^{n} r^{1+2+3+\dots+(n-1)}

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

a^{n} \left[\left(\frac{b}{a}\right)^{\frac{1}{n-1}}\right]^{\frac{n(n-1)}{2}}

a^{n} \left(\frac{b}{a}\right)^{\frac{n}{2}}

a^{\frac{n}{2}} b^{\frac{n}{2}}

(ab)^{\frac{n}{2}}

∴

P^2

\left[(ab)^{\frac{n}{2}}\right]^{2}

(ab)^n

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