(In)Equality

Algebra Level 3

For any positive real number $a$ and for any $n\in N$ , the greatest value of the given expression is- $\frac{a^{n}}{1+a+a^{2}+ a^3+\cdots + a^{2n}}$

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\frac{1}{2n-1}

\frac{1}{2n+1}

\frac{1}{2(n+1)}

\frac{1}{2n}

1 solution

Vatsalya Tandon
Jan 21, 2016

We have,

$\frac{a^{n}}{1+a+a^{2}+ a^3+\dots a^{2n}}$

$=\frac{1}{(\frac{1}{a^{n}}+\frac{1}{a^{n-1}} + \dots + \frac{1}{a}) + 1 + (a+ a^{2}+ \dots a^{n})}$

$=\frac{1}{1 + (a + \frac{1}{a})+ (a^{2} + \frac{1}{a^{2}}) + \dots + (a^{n} + \frac{1}{a^{n}})}$

$\left ( \because a^{k} + \frac{1}{a^{k}} \geqslant 2 \right )$ for $k=1,2, \dots , n$

$\leqslant \frac{1}{1+2+2+ \dots 2}$ n-times

$=\frac{1}{2n+1}$

Direct application of AM-GM will also work.

Department 8 - 5 years, 4 months ago

The third last line is also an application of AM-GM

Vatsalya Tandon - 5 years, 4 months ago

Yes! I see that but I was talking at the starting too will work.

Department 8 - 5 years, 4 months ago