Definite Integration
Reduction Formula
nta_pyq_2024_apr
Grade 12
Question:
Let $r_k=\dfrac{\int_0^1(1-x^7)^k\,dx}{\int_0^1(1-x^7)^{k+1}\,dx}$, $k\in\mathbb{N}$. Then the value of $\displaystyle\sum_{k=1}^{10}\dfrac{1}{7(r_k-1)}$ is equal to ________.
Step-by-Step Solution
Key Concept: Using integration by parts on $I_k=\int_0^1(1-x^7)^k dx$: $\frac{I_k}{I_{k+1}}=\frac{7k+8}{7k+7}$. So $r_k=\frac{7k+8}{7k+7}$, $r_k-1=\frac{1}{7(k+1)}$, $\frac{1}{7(r_k-1)}=k+1$.
$\frac{1}{7(r_k-1)}=k+1$. $\sum_{k=1}^{10}(k+1)=65$.
Correct Answer: 65