Final answer:
To find the coefficient of the third term in the expansion of (2x-1)^6, we use the binomial theorem formula to calculate it as 6! / (2! × (6-2)!) multiplied by (2x)^4. This results in a coefficient of 240 for the third term, which is represented by the term 240x^4.
Step-by-step explanation:
To find the coefficient of the third term of the binomial expansion of (2x-1)^6, we can use the binomial theorem. The general term in a binomial expansion (a+b)^n is given by the formula: T(k+1) = C(n, k) × a^(n-k) × b^k, where T(k+1) is the (k+1)-th term, C(n, k) is the binomial coefficient, a and b are the terms in the binomial, n is the exponent, and k is the specific term number - 1.
For the third term of (2x-1)^6, k would be 2 (since we start counting from 0). Therefore:
- C(6, 2) = 6! / (2! × (6-2)!) = 15
- The term will have (2x)^(6-2) = (2x)^4
- And (-1)^2 = 1
So, the third term is given by the coefficient 15 multiplied by (2x)^4, which simplifies to:
Therefore, the coefficient of the third term is 240.