Let n ∈ N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0 n , 3 · C 1 n , 5 · C 2 n , 7 · C 3 n , … is equal to 2 100 · 101 , then 2 n - 1 2 is equal to

Question

Let n &#8712; N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0 n , 3 &#183; C 1 n , 5 &#183; C 2 n , 7 &#183; C 3 n , &#8230; is equal to 2 100 &#183; 101 , then 2 n - 1 2 is equal to

Aicharya · Accepted Answer

Let S = C 0 n + 3 C 1 n + 5 . C 2 n + &#8230; + 2 n + 1 &#183; C n n Now, T r = 2 r + 1 n C r Therefore, S = &#931; T r S = &#931; 2 r + 1 n C r = &#931; 2 r n C r + &#931; n C r &#8658; S = 2 n &#183; 2 n - 1 + 2 n = 2 n n + 1 &#8658; 2 n n + 1 = 2 100 . 101 &#8658; n = 100 Now, 2 n - 1 2 = 2 99 2 = 98 .

Let n ∈ N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0…

Solution

Let n &#8712; N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0…

Solution

Let n ∈ N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0…