1. Prove that for any positive integer n, the largest value of 2nCr for 0< r < 2n (r integers) is 2nCn and that it occurs only when r = n.
2. By considering the value (1+x)^2n when x = 1, prove that ∑_(r=0)^n▒(2n¦r) = 2^(2n-1)+(2n)!/(2〖(n!)〗^2 )
3. By integrating both sides of the expansion of (1+x)^n with respect to x between suitable limits, show that ∑_(k=0)^n▒〖1/(k+1) (n¦r) 〗= (2^(n+1)-1)/(n+1)
4.
(a) find the sum of the geometric series 1 +(1+x)+(1+x)^2+...+(1+x)^n
(b) by considering the coëfficiënt of X^r, prove that (n¦r)+ ((n-1)¦r)+((n-2)¦r)+⋯(r¦r)=((n+1)¦(r+1))
can do part 1 but not part b.
please help. thanks
2. By considering the value (1+x)^2n when x = 1, prove that ∑_(r=0)^n▒(2n¦r) = 2^(2n-1)+(2n)!/(2〖(n!)〗^2 )
3. By integrating both sides of the expansion of (1+x)^n with respect to x between suitable limits, show that ∑_(k=0)^n▒〖1/(k+1) (n¦r) 〗= (2^(n+1)-1)/(n+1)
4.
(a) find the sum of the geometric series 1 +(1+x)+(1+x)^2+...+(1+x)^n
(b) by considering the coëfficiënt of X^r, prove that (n¦r)+ ((n-1)¦r)+((n-2)¦r)+⋯(r¦r)=((n+1)¦(r+1))
can do part 1 but not part b.
please help. thanks