Final answer:
To find the value of n for which the coefficient of x² is 90 in the expansion of (1−3x)ⁿ, the binomial theorem is used. The coefficient of x² is found by the binomial coefficient nC2, leading to the equation n² - n - 20 = 0, which has a positive solution of n = 5.
Step-by-step explanation:
To determine the value of n for which the coefficient of x² is 90 in the expansion of (1−3x)ⁿ, we use the binomial theorem. The binomial theorem gives us a way to expand expressions raised to a power and is written as:
- (a + b)ⁿ = aⁿ + naⁿ−1b + ⅟n(n-1)aⁿ−62 + ⅟n(n-1)(n−2)aⁿ∓63 + ... + bⁿ
The general term in the expansion of (a + b)ⁿ is given by T(k+1) = nCk × aⁿ−k × b⁾k, where nCk is the binomial coefficient.
To find the coefficient of x², we need the term where k=2. In our case, a = 1 and b = -3x, and thus the term becomes:
- T(3) = nC2 × 1ⁿ−2 × (-3x)² = n(n-1)/2! × 9x²
So, the coefficient of x² is:
Simplifying gives us:
- 9n(n - 1) = 180
- n(n - 1) = 20
- n² - n - 20 = 0
- (n - 5)(n + 4) = 0
Since n is positive, n = 5 is the value of n we're looking for.