Determine the 2nd term in the binomial expansion of (a + 3b$$)^4 .(C:1)

Question

BlackTom AI · Accepted Answer

To find the second term in the expansion of (a + 3b)^4, recall the binomial expansion formula: the k-th term (starting from k = 0) is C(n, k) a^{n-k} (3b)^k with n = 4.

First, identify which term is the second one. In binomial expansion indexing, the second term corresponds to k = 1.

Compute the k = 1 term: C(4, 1) * a^{4-1} * (3b)^1 = 4 * a^3 * 3b.

Carrying out the multiplication gives 4 * 3 * a^3 b = 12 a^3 b.

Now evaluate each option:
- Option a: 12 a^3 b — this matches the computed second term exactly.
- Option b: a^4 b^4 — this would correspond to a different combination of exponents and does not arise in this expansion for any k.
- Option c: a^3 b^4 — the exponents do not align with any term from the expansion of (a + 3b)^4; the powers of a and b here are mismatched.
- Option d: 3ab — this would be far too small in power and does not reflect the binomial coefficients or the required a^3 factor for the second term.

The correct second term is the one that preserves a^3 and introduces a single b with the appropriate coefficient, which is 12 a^3 b.

Determine the 2nd term in the binomial expansion of (a + 3b$$)^4 .(C:1)

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