 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/9/solution

(1 + a x)^{n} = 1 + na x + \frac{n ( n - 1 )}{2} a^{2} x^{2} + O (x^{3}) .

na = - 2

and

n (n - 1) a^{2} /2 = 3

. Since

na \neq = 0

, substitute

a = - 2/ n

into the second equation to obtain

2 (n - 1) / n = 3

, so

n = - 2

and

a = 1

. Therefore

a = 1, n = - 2.

Indeed

(1 + x)^{- 2} = 1 - 2 x + 3 x^{2} - \dots

for

∣ x ∣ < 1

 New to topics? Read the docs here!