Odd And Even Coefficient Sums

Question

If the expansion of (1 + x)^n is split into terms with even index and terms with odd index, the sum of the even-indexed coefficients separately equals which value?

RankGuru · Accepted Answer

Separating even and odd indexed coefficients is achieved by combining the substitutions x = 1 and x = -1, a paired-evaluation technique that JEE Advanced uses to split sums. Let E = C(n,0) + C(n,2) + C(n,4) + ... be the even-index sum and O = C(n,1) + C(n,3) + ... be the odd-index sum. Substituting x = 1 in (1 + x)^n gives E + O = 2^n, while substituting x = -1 gives E - O = 0 since (1 - 1)^n = 0 for positive n. Adding these two equations yields 2E = 2^n, so E = 2^{n-1}; subtracting yields the same for O. Hence the even-indexed coefficient sum is 2^{n-1}. Option 2^n is the total of all coefficients, not just the even-indexed half. Option 2^{n+1} doubles the full sum erroneously. Option n^2 has no connection to the binomial structure. The symmetry E = O = 2^{n-1} reflects the equal numbers of even-sized and odd-sized subsets of an n-element set. Plausibility check: for n = 3, the even-index coefficients 1 + 3 = 4 = 2^{2}, exactly 2^{n-1}, confirming the result.

Odd And Even Coefficient Sums

Select the correct option:

🔒 Solution Hidden from View

More binomial theorem and its simple applications Practice Questions

When the trinomial (1 + x + x^2)^{3} is expanded completely into a polynomial in x, the coefficient ...

Using the first-order binomial approximation for small quantities, the approximate value of (1.002)^...

Treating the binomial theorem for a negative index, the coefficient of x^3 in the series expansion o...

Within the expansion of (\sqrt{3} + \sqrt[3]{2})^{6}, the number of terms that turn out to be ration...

By differentiating a standard expansion and substituting suitably, the weighted sum C(n,1) + 2 C(n,2...

Through integrating the binomial expansion over a unit interval, the alternating series C(n,0)/1 - C...

Suppose the coefficients of three consecutive terms in the expansion of (1 + x)^n are in the ratio 1...

For the numerical expansion of (1 + x)^{n} with specific values applied, the greatest term is identi...

Using the binomial theorem to analyse remainders, the quantity 7^{83} is divided by 6 and the result...

When the very large number 3^{100} is computed and its final two digits are examined, the binomial t...

A question requires the coefficient of x^5 in the product expansion of (1 + x)^4 multiplied by (1 + ...

Evaluating the alternating combination C(n,0) - C(n,1) + C(n,2) - C(n,3) + ... over a full binomial ...

About This Question

Solution