Question

Trinomial Expansion: Find the value of "c" that satisfies the given condition.
Problem posted below.

Answer 1

\[\Large (3x^2 + 2x + c)^{12} = \sum_{r = 0}^{24}A_rx^r \\ \Large \frac{ A_{19} }{ A_5 } = \frac{1}{2^7}\]

Answer 2

Find the value of "c".

I solved it the long way using Binomial Theorem and selectively finding the coefficients of x^19 and x^5 and found all values of "c".

But is there a quick way to find just one value of c that will satisfy the given condition? (Because I am told this problem was in a math test that was an hour long and there were 20 questions total giving an average of about 3 minutes per problem).

If I put x = 0, I get A0 = c^12. How to find the ratio A19/A5 quickly? (RHS will have 25 terms. I noticed A19 is the 6th term from the end and A5 is the sixth term from the beginning if that mattered.)