Question

In the binomial expression (ax+b) ^20 where

a and b are positive, the coefficient of x^k−1
is twice the coefficient of x^k.
a) True
b) False

Answer 1

In the binomial expansion (ax+b)^20, the coefficient of x^(k-1) is not twice the coefficient of x^k. It depends on the binomial coefficient and the exponent of x.

Step-by-step explanation:

The expression in the binomial expansion is (ax+b)^20. We need to find the coefficient of x^(k-1). In the binomial expansion, the coefficient of x^k is given by the binomial coefficient C(20, k). The coefficient of x^(k-1) is then obtained by multiplying the coefficient of x^k by the exponent of x, which is k. Therefore, the coefficient of x^(k-1) is C(20, k) * k.

Let's compare the coefficient of x^(k-1) to the coefficient of x^k. If the coefficient of x^(k-1) is twice the coefficient of x^k, then C(20, k) * k = 2 * C(20, k+1). By simplifying the expression, we get k = 2 * (20 - k - 1). Solving for k, we find that k = 18.

So, the coefficient of x^17 is twice the coefficient of x^18 in the given binomial expansion. Therefore, the statement is False.