Question:
If the 21st and 22nd terms in the expansion of $(1 + x)^{44}$, are equal, then x =
If the 21st and 22nd terms in the expansion of $(1 + x)^{44}$, are equal, then x =
Updated On: May 25, 2024
- $\frac {7} {8}$
- $\frac {8} {7}$
- $\frac {21} {22}$
- $\frac {23} {24}$
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The Correct Option is A
Solution and Explanation
$T_{21} = T_{20+1} = \,^{44}c_{20}x^{20}$
$T_{22} = T_{21+1} = \,^{44}c_{21}\left(-x\right)^{21} = - \,^{44}c_{21}x^{21}$
$\therefore \,^{44}c_{20}x^{20} = - \,^{44}c_{21}x^{21}$
$\therefore x = \frac{^{44}c_{20}}{^{44}c_{21}} = \frac{44\,!}{24\,!\,20\,!} \times \frac{21\,!\,3!}{44\,!}$
$= \frac{21\cdot20\,!\,23\,!}{20\,!\,24\cdot23\,!} = -\frac{21}{24} = -\frac{7}{8}$
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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.