Expansion of (1 + x)^60

To solve this problem, we need to expand the expression (1 + x)^60 and find the sum of coefficients of odd powers of x.

The general formula for expanding the binomial expression (a + b)^n is given by the binomial theorem:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

Where C(n, r) represents the binomial coefficient, given by:

C(n, r) = n! / (r! * (n-r)!)

In this case, we have (1 + x)^60, so a = 1, b = x, and n = 60.

Sum of coefficients of odd powers of x

We want to find the sum of coefficients of odd powers of x. This means we need to consider the terms where the exponent of x is an odd number.

Since the exponent of x in each term is given by n - r, we need to consider the terms where (n - r) is odd.

For an even number n, when (n - r) is odd, it means that r is even. Similarly, when (n - r) is even, it means that r is odd.

Therefore, the sum of coefficients of odd powers of x is given by the sum of coefficients where r is even.

Using the binomial theorem

Using the binomial theorem, we can expand (1 + x)^60:

(1 + x)^60 = C(60, 0) * 1^60 * x^0 + C(60, 1) * 1^59 * x^1 + C(60, 2) * 1^58 * x^2 + ... + C(60, 58) * 1^2 * x^58 + C(60, 59) * 1^1 * x^59 + C(60, 60) * 1^0 * x^60

Calculating the sum of coefficients of odd powers of x

We only need to consider the terms where r is even, which means we need to consider terms with r = 0, 2, 4, ..., 58, 60.

The terms with even values of r are:

C(60, 0) * 1^60 * x^0 + C(60, 2) * 1^58 * x^2 + C(60, 4) * 1^56 * x^4 + ... + C(60, 58) * 1^2 * x^58 + C(60, 60) * 1^0 * x^60

We can see that the terms with even values of r are symmetrical. For every term with an even value of r, there is a corresponding term with the same coefficient but with a negative exponent of x.

Therefore, when we add up all the terms with even values of r,