Question:
r and n are positive integers $r > 1, n > 2$ and coefficient of $(r+2)^{th}$ term and $3r^{th}$ term in the expansion of $(1 + x)^{2n }$ are equal, then n equals
r and n are positive integers $r > 1, n > 2$ and coefficient of $(r+2)^{th}$ term and $3r^{th}$ term in the expansion of $(1 + x)^{2n }$ are equal, then n equals
Updated On: Jul 5, 2022
- $3r$
- $3r + 1 $
- $2r$
- $2r + 1 $
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The Correct Option is C
Solution and Explanation
$t_{r+2} = {^{2n}C_{r+1}} \,x^{r+1}; t_{3r} = {^{2n}C_{3r-1} } x^{3r-1}$
Given $ {^{2n}C_{r+1} } = {^{2n}C_{3r-1}} ; $
$\Rightarrow \, {^{2n }C_{2n -\left(r+1\right)}} = {^{2n }C_{3r-1}} $
$\Rightarrow 2n-r-1=3r-1 \Rightarrow 2n = 4r \Rightarrow n= 2r $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.