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State true or false. For every integer n> 1, the inequality {

Question

State true or false.
For every integer $n > 1$ , the inequality $(n!)^{1/ n} < \frac{n + 1}{2}$ holds.

True

False

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Emily Discussed

State true or false.

For every integer

n > 1

, the inequality

(n!)^{1/ n} < \frac{n + 1}{2}

holds.

14 mins ago

Discuss this question LIVE

14 mins ago

Text solutionVerified

Consider

n

numbers, namely

1, 2, 3, 4, ..... n

. Now using

A . M > G . M

for distinct numbers we get

\frac{1 + 2 + 3 + .... + n}{n} > (1 \times 2 \times 3 \times 4 \times .... \times n)^{1/ n}

\Rightarrow \frac{n ( n + 1 )}{2 n} > (n!)^{1/ n}

\Rightarrow (n!)^{1/ n} < \frac{( n + 1 )}{2}

Hence, the statement is true

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Question Text	State true or false. For every integer $n > 1$ , the inequality $(n!)^{1/ n} < \frac{n + 1}{2}$ holds.
Answer Type	Text solution:1
Upvotes	150