Let r_k= ∫_0^1(1-x^7)^k d x ∫_0^1(1-x^7)^ k+1 d x , k ∈ ℕ . Then the value of ∑_ k=1 ^ 10 (1)/(7(r_k-1)) is equal to________

Correct answer: 0 aligned & I_K=∫ 1 ·(1-x^7)^K d x & I_K=.(1-x^7)^K x|_0 ^1+7 K ∫_0^1(1-x^7)^ K-1 x^6 · x d x & I_K=-7 K ∫_0^1(1-x^7)^ K-1 ((1-x^7)-1) d x & I_K=-7 K I_K+7 K I_ K-1 & I_K I_ K+1 =(7 K+8)/(7 K+7) aligned aligned & r_K= 7 K+8 7 ~K+7 & r_ K -1= 1 7(~K+1) & 7(r_ K -1)= 1 ~K+1 & ∑_ K=1 ^ 10 (~K+1)=11(6)-1=65 aligned

JEE Main 20246 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Definite Integration

Let r_k= ∫_0^1(1-x^7)^k d x ∫_0^1(1-x^7)^ k+1 d x , k ∈ ℕ . Then...

Let

r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}

. Then the value of

\sum_{k=1}^{10} \frac{1}{7\left(r_k-1\right)}

is equal to________

Correct answer

Step-by-step solution

\begin{aligned} & I_K=\int 1 \cdot\left(1-x^7\right)^K d x \\ & I_K=\left.\left(1-x^7\right)^K x\right|_0 ^1+7 K \int_0^1\left(1-x^7\right)^{K-1} x^6 \cdot x d x \\ & I_K=-7 K \int_0^1\left(1-x^7\right)^{K-1}\left(\left(1-x^7\right)-1\right) d x \\ & I_K=-7 K I_K+7 K I_{K-1} \\ & \Rightarrow \frac{I_K}{I_{K+1}}=\frac{7 K+8}{7 K+7}\end{aligned}

\begin{aligned} & r_K=\frac{7 K+8}{7 \mathrm{~K}+7} \\ & \mathrm{r}_{\mathrm{K}}-1=\frac{1}{7(\mathrm{~K}+1)} \\ & \Rightarrow 7\left(\mathrm{r}_{\mathrm{K}}-1\right)=\frac{1}{\mathrm{~K}+1} \\ & \sum_{\mathrm{K}=1}^{10}(\mathrm{~K}+1)=11(6)-1=65\end{aligned}

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