Find the term indicated in each expansion. (2x + y)⁶; third term

Verified step by step guidance

1. Identify the general term formula for the binomial expansion of \((a + b)^n\), which is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\), where \(k\) starts from 0.
2. In this problem, \(a = 2x\), \(b = y\), and \(n = 6\). We are asked to find the third term, so set \(k = 2\) because the term number is \(k+1\).
3. Calculate the binomial coefficient \(\binom{6}{2}\), which represents the number of ways to choose 2 elements from 6.
4. Substitute the values into the general term formula: \(T_3 = \binom{6}{2} (2x)^{6-2} (y)^2\).
5. Simplify the powers and coefficients as much as possible to express the third term in expanded form (without multiplying out the numbers).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n. It states that the expansion is the sum of terms involving binomial coefficients multiplied by powers of a and b. This theorem helps find any specific term in the expansion without fully expanding the expression.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal’s Triangle. These coefficients determine the weight of each term in the expansion.

General Term in Binomial Expansion

The general term (k+1) in the expansion of (a + b)^n is given by T(k+1) = C(n, k) * a^(n-k) * b^k. This formula allows direct calculation of any term by substituting the values of n, k, a, and b. For example, the third term corresponds to k = 2.