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An Important Theorem - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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QuestionMEDIUM

\text { If } \frac{1}{n+1}{ }^n \mathrm{C}_{\mathrm{n}}+\frac{1}{n}{ }^n \mathrm{C}_{\mathrm{n}-1}+\ldots+\frac{1}{2}{ }^n \mathrm{C}_1+{ }^n \mathrm{C}_0=\frac{1023}{10} \text { then } n \text { is equal to }

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Concepts Covered - 1

An Important Theorem

Finding nature of an integral part of the expression.

If the given expansion is in the form of \mathrm{N}=(\mathrm{a}+\sqrt{\mathrm{b}})^{\mathrm{n}} \quad(\mathrm{n} \in \mathrm{N})

Working rule:

Step 1: \mathrm{Choose\;\;N}^{\prime}=(\mathrm{a}-\sqrt{\mathrm{b}})^{\mathrm{n}} \text { or }(\sqrt{\mathrm{b}}-\mathrm{a})^{\mathrm{n}} \text { according as a }>\sqrt{\mathrm{b}} \text { or } \sqrt{\mathrm{b}}>\mathrm{a}

 

Step 2: Use N + N’ or N - N’ such that result is an integer

I.e. \mathrm{(a+\sqrt{b})^n+(a-\sqrt{b})^n\;\;or\;\;(a+\sqrt{b})^n-(a-\sqrt{b})^n\;\;is\;\;an\;integer}

 

Step 3: Now use the concept greatest integer function and fractional part of a function, N = I + f, where I is an integral part of N i.e., [N] and f is a fractional part of N, i.e. { N }.


 

For example, the integral part of P=(3 \sqrt{3}+5)^{2 n+1}(n \in N) is an even number.

Now consider, P^{\prime}=(3 \sqrt{3}-5)^{2 n+1} \text { here } 0<P^{\prime}<1

Use, P-P^{\prime}=2\left[^{2 n+1} C_{1}(3 \sqrt{3})^{2 n} 5^{1}+^{2 n+1} C_{3}(3 \sqrt{3})^{2 n-2}(5)^{3}+\ldots \ldots .\right]

\begin{array}{l}{I+f-P^{\prime}=2 k(k \in N)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{ (P=I+f )} \\\\ {-1<f-P^{\prime}<1 \;\mathrm { but }\; f-P^{\prime} \text { is an integer } \Rightarrow f-P^{\prime}=0 \Rightarrow I=2 k}\end{array}

Hence, integral part of P=(3 \sqrt{3}+5)^{2 n+1}(n \in N)  is an even integer

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An Important Theorem

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General and Middle Terms
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An Important Theorem
Problems on Divisibility
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Multinomial Theorem
Series Involving Binomial Coefficients

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