1.1+3+5+.......+(2n-1)=n^(2)

Prove that ((2n+1)!)/(n !)=2^n{1. 3. 5 .........(2n-1)(2n+1)}

By the Principle of Mathematical Induction, prove the following for all n in N : 1+3+5+....... +(2n-1) =n^2 i.e. the sum of first » odd natural numbers is n^2 .

Using mathematical induction prove that P(n) = 1+3+5+........... +2n-1=n^(2)

For the proposition P(n), given by , 1+3+5+.........+(2n-1) = n^2 +2 , prove that P(k) is true implies P(k + 1) is true. But, P(n) is not true for all n in N.

For all ninNN , prove by principle of mathematical induction that, 1+3+5+ . . .+(2n-1)=n^(2) .

Show that the middle term in the expansion of (x+1)^(2n)" is " (1.3.5. ......(2n-1))/(n!).2^(n).x^(n).

Prove that the term independent of x in the expansion of (x+1/x)^(2n) \ is \ (1. 3. 5 ... (2n-1))/(n !). 2^n .

Show that the middle term in the expansion of (1 + x)^(2n) is (1.3.5.........(2n - 1))/(n!)2^(n)x^(n) , where n is a positive integer.

Show that the middle term in the expansion of (x-1/x)^(2n) is (1.3.5.7....(2n-1))/(n!)(-2)^n

Prove that 1+3+5+7......+(2n+1)=(n+1)^(2)