If the expansion of (x + a)^n has a term 15x^3a^2, what is the value of n?

If the expansion of (x + a)^n has a term 15x^3a^2, what is the value of n?

Step 1: Understand that the expression (x + a)^n can be expanded using the binomial theorem.
Step 2: The general term in the expansion of (x + a)^n is given by C(n, k) * x^(n-k) * a^k, where C(n, k) is the binomial coefficient.
Step 3: Identify the term we are interested in: 15x^3a^2. Here, x has an exponent of 3 and a has an exponent of 2.
Step 4: From the term 15x^3a^2, we can see that k (the exponent of a) is 2, and n - k (the exponent of x) is 3.
Step 5: Set up the equation: n - k = 3, which means n - 2 = 3. Therefore, n = 3 + 2.
Step 6: Calculate n: n = 5.
Step 7: Now, we need to find the binomial coefficient C(n, 2) and set it equal to 15. C(n, 2) = n! / (2!(n-2)!).
Step 8: Substitute n = 5 into C(n, 2): C(5, 2) = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10.
Step 9: Since we need C(n, 2) * a^2 = 15, we can see that we need to find n such that C(n, 2) = 15.
Step 10: Solve for n: C(n, 2) = n(n-1)/2 = 15, which leads to n(n-1) = 30.
Step 11: Solve the quadratic equation n^2 - n - 30 = 0. The solutions are n = 6 or n = -5.
Step 12: Since n must be a positive integer, we conclude that n = 6.

Binomial Expansion – Understanding how to expand expressions of the form (x + a)^n and identify specific terms using the binomial theorem.
Binomial Coefficient – Using the binomial coefficient C(n, k) to determine the coefficients of terms in the expansion.
Identifying Terms – Recognizing the structure of terms in the expansion to extract necessary information for solving for n.