Given positive integers $r >1, n > 2$ and the coefficient of $(3r)th$ and $ (r + 2)th $ terms in the binomial expansion of $(1 + x)^2n$ are equal. Then,
- n=2r
- n=2r+1
- n=3r
- None of these
The Correct Option is A
Solution and Explanation
and $\hspace20mm \, t_{r+2}= \, ^{2n}C_{r+1} (x) ^{r+1}$
Since, binomial coefficients of t$_3r$ and t$_{r+ 2}$ are equal
$\Rightarrow \, \, \, \, \, \, \, \, 3r-1=r+1 \, or \, 2n=(3r-1)+(r+1)$
$\Rightarrow \, \, \, \, \, \, \, \, $ 2r=2 $ \, or \, \, \, \, \, \, \, \, $ 2n = 4r
$\Rightarrow \, \, \, \, \, \, \, \, $ r = 1 $ \, or \, \, \, \, \, \, \, \, $ n=2r
But $ \, \, \, \, \, \, \, \, $ r >1
$\therefore \, \, $We take, 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.