Let n∈N and x denote the greatest integer less than or equal to x. If the sum of n+1 terms of C0n,3·C1n,5·C2n,7·C3n,… is equal to 2^100·101, then…

Question

Let n∈N and x denote the greatest integer less than or equal to x. If the sum of n+1 terms of C0n,3·C1n,5·C2n,7·C3n,… is equal to 2^100·101, then 2(n-1)/2 is equal to

MARKS Content Team · Accepted Answer

Correct answer: 98. Let S=C0n+3C1n+5.C2n+…+2n+1·Cnn Now, T_r=2r+1^nC_r Therefore, S=ΣT_r S=Σ2r+1^nC_r =Σ2r^nC_r+Σ^nC_r ⇒S=2n·2^(n-1)+2^n=2^nn+1 ⇒2^nn+1=2^100.101 ⇒n=100 Now, 2(n-1)/2=299/2=98.

Let $n \in N$ and $[x]$ denote the greatest integer less than or equal to $x .$ If the sum of $(n + 1)$ terms of ${}^{n}C_{0}, 3 \cdot {}^{n}C_{1}, 5 \cdot {}^{n}C_{2}, 7 \cdot {}^{n}C_{3}, \dots$ is equal to $2^{100} \cdot 101,$ then $2 [\frac{n - 1}{2}]$ is equal to

Solution

Relevant Questions

Let n∈N and x denote the greatest integer less than or equal to x. If the sum of n+1 terms of C0n,3·C1n,5·C2n,7·C3n,… is equal to 2100·101, then 2n-12 is equal to

Solution

Relevant Questions

Let $n \in N$ and $[x]$ denote the greatest integer less than or equal to $x .$ If the sum of $(n + 1)$ terms of ${}^{n}C_{0}, 3 \cdot {}^{n}C_{1}, 5 \cdot {}^{n}C_{2}, 7 \cdot {}^{n}C_{3}, \dots$ is equal to $2^{100} \cdot 101,$ then $2 [\frac{n - 1}{2}]$ is equal to