Question:
The value of $\frac{1}{81^n}-\frac{10}{81^n}\,^{2n}C_1+\frac{10^2}{81^n}\,^{2n}C_2-\frac{10^3}{81^n}\,^{2n}C_3+.....+\frac{10^{2n}}{81^n}is$
The value of $\frac{1}{81^n}-\frac{10}{81^n}\,^{2n}C_1+\frac{10^2}{81^n}\,^{2n}C_2-\frac{10^3}{81^n}\,^{2n}C_3+.....+\frac{10^{2n}}{81^n}is$
Updated On: Jul 7, 2022
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- $\frac{1}{2}$
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The Correct Option is D
Solution and Explanation
The given expression
$= \frac{1}{81^{n}}[^{2n}C_{0}-\,^{2n}C_{1}\,10^{1}+\,^{2n}C_{2}\,10^{2}$
$-\,^{2n}C_{3}\,10^{3}+.... +\,^{2n}C_{2n}\,10^{2n}]$
$= \frac{1}{81^{n}}\left[1-10\right]^{2n}= \frac{\left(-9\right)^{2n}}{81^{n}} = \frac{81^{n}}{81^{n}}=1$
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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.