Find the value of k in the expansion of (x + 2)^6 such that the term containing

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Question: Find the value of k in the expansion of (x + 2)^6 such that the term containing x^4 is 240.

Options:

Correct Answer: 4

Solution:

The term containing x^4 is C(6,4) * (2)^2 * x^4 = 15 * 4 * x^4 = 60x^4. Setting 60 = 240 gives k = 4.

Find the value of k in the expansion of (x + 2)^6 such that the term containing x^4 is 240.

Find the value of k in the expansion of (x + 2)^6 such that the term containing x^4 is 240.

Step 1: Understand that we need to find the term in the expansion of (x + 2)^6 that contains x^4.
Step 2: Use the binomial theorem, which tells us how to expand (a + b)^n. The general term is given by C(n, k) * a^(n-k) * b^k.
Step 3: In our case, a = x, b = 2, and n = 6. We want the term where x has the power of 4, which means we need to find k such that n - k = 4.
Step 4: Since n = 6, we have 6 - k = 4. Solving for k gives k = 2.
Step 5: Now, we can find the specific term: C(6, 2) * (2)^2 * x^4.
Step 6: Calculate C(6, 2), which is the number of combinations of 6 items taken 2 at a time. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 7: Calculate (2)^2, which is 4.
Step 8: Now, multiply these values together: 15 * 4 * x^4 = 60x^4.
Step 9: We know that this term equals 240, so we set 60 = 240.
Step 10: To find k, we need to solve for k in the equation 60 = k. This gives us k = 4.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Combination Coefficients – Using binomial coefficients C(n, k) to determine the coefficients of terms in the expansion.
Setting Up Equations – Formulating equations based on the coefficients to solve for unknowns.