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  4. Important Result (Comparison) - Practice Questions & MCQ

Important Result (Comparison) - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

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  • 4 Questions around this concept.

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QuestionMEDIUM

If y=(x)^{3x} for x>2, which of the following is true?

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The value of the natural numbers n such that the inequality  2^n>2 n+1  is valid:

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Concepts Covered - 1

Important Result (Comparison)

Important Result (Comparison)

    2\leq\left ( 1+\frac{1}{n} \right )^n<3,\;\;\;\;\;n\in\mathbb{N}

Proof:

Expand, \\ \left ( 1+\frac{1}{n} \right )^n \text{using binomial theorem} \\\\

\begin{aligned}\left(1+\frac{1}{n}\right)^{n} &=1+n \frac{1}{n}+\frac{n(n-1)}{2 !} \frac{1}{n^{2}}+\frac{n(n-1)(n-2)}{3 !} \frac{1}{n^{3}}+\cdots+\frac{n(n-1)(n-2) \cdots[n-(n-1)]}{n !} \frac{1}{n^{n}} \\&={1+1+\frac{1}{2 !}\left(1-\frac{1}{n}\right)+\frac{1}{3 !}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)} {+\cdots+\frac{1}{n !}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \dots\left(1-\frac{n-1}{n}\right)}\\&{<1+1+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !}}\\&{<1+1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots+\frac{1}{2^{n-1}}}=1+1 \frac{\left\{1-\left(\frac{1}{2}\right)^{n}\right\}}{1-\frac{1}{2}}=1+2\left\{1-\left(\frac{1}{2}\right)^{n}\right\}=3-\frac{1}{2^{n-1}}\end{aligned}

Hence, from above

2\leq\left ( 1+\frac{1}{n} \right )^n<3,\;\;n\geq1

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Important Result (Comparison)

Study other Related Concepts

Important Result (Comparison) Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

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Important Result (Comparison)

Mathematics Textbook for Class VII

Page No. : 8.4

Line : 14

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