`1+2.2 +3.2^2 + 4.2^3` + … `t_n` is :

Prove the following by using the principle of mathematical induction for all n in N : 1. 2. 3 + 2. 3. 4 + .. . + n(n + 1) (n + 2)=(n(n+1)(n+2)(n+3))/4

The sum to 100 terms of the series 1.2.3. + 2.3.4. + 3.4.5. + …+ n(n + 1) (n+2) + …. is integral multiple of

Using the principle of mathematical induction, prove that 1.3 + 2.3^(2) + 3.3^(2) + ... + n.3^(n) = ((2n-1)(3)^(n+1)+3)/(4) for all n in N .

Mean of the numbers 1,2,3,…,n with respective weights 1^(2) + 1, 2^(2) + 2, 3^(2) + 3,…,n^(2)+n is

Find the 1+2.2+3.2^2+…….+t_n

If (1 ^(2) - t _(1)) + (2 ^(2) - t _(2)) + ......+ ( n ^(2) - t _(n)) =(1)/(3) n ( n ^(2) -1 ), then t _(n) is

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :

If t a n1^0=t , the value of "cos"2^0+"t s i n"2^0 is t 2. 1 3. 1/2 4. 0 5. -1

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + c_(3) x^(3) + …+ C_(n) x^(n) , show that sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4)) = (1)/((n+1)(n+2)(n+3)(n+4))(4^(n+4) -sum_(t=0)^(3) ""^(n+4)C_(t)) .

Find the area of the triangle whose vertices are (6,\ 3),\ (-3,\ 5) and (4,\ -2) (ii) (a t1 2,\ 2a t_1),\ (a t2 2,\ 2a t_2) and (a t3 2,\ 2a t_3) (iii) (a ,\ c+a),\ (a ,\ c) and (-a ,\ c-a)