The coefficient of y⁴ in the expansion of (2y - 5)⁷ is calculated using the binomial theorem and is found to be 175000.
To find the coefficient of y⁴ in the expansion of (2y - 5)⁷, we can use the binomial theorem. The general term in the expansion of (a + b)⁷ is given by T(k+1) = C(n, k) · a⁷⁻⁻· b⁻⁻, where C(n, k) is the binomial coefficient, and k is the term number minus 1.
For the coefficient of y⁴, k will be equal to 4. Thus, we need the fifth term in the expansion where a = 2y and b = -5:
T(5) = C(7, 4) · (2y)³ · (-5)⁴
Now, we simply need to calculate this term:
- C(7, 4) = 35
- (2y)³ = 8y³
- (-5)⁴ = 625
Multiplying these together gives the fifth term containing y⁴:
T(5) = 35 · 8y⁴ · 625 = 175000y⁴
Thus, the coefficient of y⁴ in the expansion is 175000.