Given n4 - 10n for a fixed positive integer n - 2- then n-14 - 10n-1

We are given n^4 lt; 10^n Also for n ≥ 2, we have ( 1+1/n )^4 = ( 3/2 )^4 lt; 10 ∴ From (1) and (2), we get n^4 ( 1+1/n )^4 lt;10^n. 10 or (n+1)^4 l ...

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Standard XIII

Mathematics

Inequality

Given n4 < 10...

Question

Given $n^{4} < 10^{n}$ for a fixed positive integer $n \geq 2$ , then $(n + 1)^{4} < 10^{n + 1}$

True

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False

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Solution

The correct option is A True
We are given $n^{4} < 10^{n}$
Also for $n \geq 2$ , we have
${(1 + \frac{1}{n})}^{4} = {(\frac{3}{2})}^{4} < 10$
$∴$ From (1) and (2), we get
$n^{4} {(1 + \frac{1}{n})}^{4} < 10^{n} . 10 o r (n + 1)^{4} < 10^{n + 1}$

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Given n4<10n for a fixed positive integer n≥2, then (n+1)4<10n+1

The correct option is A TrueWe are given n4<10n Also for n≥2, we have (1+1n)4=(32)4<10 ∴ From (1) and (2), we get n4(1+1n)4<10n.10or(n+1)4<10n+1

Given $n^{4} < 10^{n}$ for a fixed positive integer $n \geq 2$ , then $(n + 1)^{4} < 10^{n + 1}$

The correct option is A True
We are given $n^{4} < 10^{n}$
Also for $n \geq 2$ , we have
${(1 + \frac{1}{n})}^{4} = {(\frac{3}{2})}^{4} < 10$
$∴$ From (1) and (2), we get
$n^{4} {(1 + \frac{1}{n})}^{4} < 10^{n} . 10 o r (n + 1)^{4} < 10^{n + 1}$