For integer ` n gt 1`, the digit at unit's place in the number
` sum_(r=0)^(100) r! + 2^(2^(n))` I

Find the sum sum_(i=0)^r.^(n_1)C_(r-i) .^(n_2)C_i .

The sum of the series sum_(r=0) ^(n) ""^(2n)C_(r), is

If x + (1)/(x) = 1 " and" p = x^(4000) + (1)/(x^(4000)) and q is the digit at unit place in the number 2^(2^(n)) + 1, n in N abd n gt 1 , then p + q is .

If n is an even natural number , then sum_(r=0)^(n) (( -1)^(r))/(""^(n)C_(r)) equals

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is even, the value of sum_(r=0)^(n//2-1) a_(2r) is

If n is an odd natural number, then sum_(r=0)^n (-1)^r/(nC_r) is equal to

lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

Find the sum sum_(r=1)^(n) r^(2) (""^(n)C_(r))/(""^(n)C_(r-1)) .

The sum of all digits of n for which sum _(r =1) ^(n ) r 2 ^(r ) = 2+2^(n+10) is :