Definite Integration
Reduction Formula — Finding $k$
nta_pyq_2024_apr
Grade 12
Question:
The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_0^1(1-x^k)^n\,dx$, $n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is:
Step-by-Step Solution
Key Concept: Reduction: $\frac{I_n}{I_{n-1}}=\frac{nk}{nk+1}$. So $\frac{I_{21}}{I_{20}}=\frac{21k}{21k+1}$. Given $\frac{I_{21}}{I_{20}}=\frac{147}{148}$.
$21k=147\Rightarrow k=7$.
Correct Answer: 4