What is the value of Σk = 028 C(28, k) · 3k ?

Concept

The binomial theorem states that for any number x and any non-negative integer n, (1 + x)n = Σk = 0n C(n, k) · xk. Read from right to left, the identity says that a sum of binomial coefficients weighted by successive powers of x, taken over the complete index range k = 0 to k = n, always collapses into the single power (1 + x)n.

Application

1. Match the given sum against that pattern. The binomial coefficient that appears is C(28, k), so the upper index is n = 28.

2. The weight multiplying each coefficient is 3k, so the variable is x = 3.

3. The index runs over the complete range k = 0 to k = 28, which is exactly the range the identity requires, so the collapse applies: Σk = 028 C(28, k) · 3k = (1 + 3)28 = 428.

4. Rewrite that result as a power of 2. Since 4 = 22, 428 = (22)28 = 22 × 28 = 256.

Cross-check

Test the same identity on a small case that can be added by hand. For n = 2 and x = 3, Σk = 02 C(2, k) · 3k = 1 + 2 × 3 + 9 = 16, and 42 = 16, so the collapse behaves exactly as the theorem predicts.

It also helps to see why the near-miss forms cannot appear:

• 329 is a pure power of 3, the shape produced by a geometric sum of powers of 3 in which no binomial coefficients appear at all.

• 3 × 227 and 3 × 427 each carry one loose factor of 3 outside the power; the binomial collapse absorbs every factor of 3 into the base (1 + 3), so no stray multiplier survives.

Result: the sum equals 256.