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Properties of Binomial Coefficients

The ratio of ...

Question

The ratio of sum of binomial coeffcients of even terms to the sum of binomial coeffcients of odd terms is

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Solution

As we know,
Sum of binomial coeffcients of even terms = Sum of binomial coeffcients of odd terms
$^{n} C_{1} +^{n} C_{3} +^{n} C_{5} + . . . . = 2^{n - 1} =^{n} C_{0} +^{n} C_{2} +^{n} C_{4} + . . . .$

Hence, $\frac{Sum of binomial coeffcients of even terms}{Sum of binomial coeffcients of odd terms} = \frac{2^{n - 1}}{2^{n - 1}} = 1$

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