Show that `(1xx2^2+2xx3^2++nxx(n1)^2)/(1^2xx2+2^2xx3++n^2xx(n+1))=(3n+5)/(3n+1)dot`

Show that (1xx2^2+2xx3^2+dotdotdot+nxx(n+1)^2)/(1^2xx2+2^2xx3+dotdotdot+n^2xx(n+1))=(3n+5)/(3n+1)dot

Show that (1xx2^2+2xx3^2+dotdotdot+nxx(n+1)^2)/(1^2xx2+2^2xx3+dotdotdot+n^2xx(n+1))=(3n+5)/(3n+1)dot

Show that (1 xx 2^(2) + 2 xx 3^(2) + . . . + n xx ( n + 1) ^(2))/( 1 ^(2) xx 2 + 2^(2) xx 3 + . . . + n^(2) xx ( n + 1) ) = ( 3 n + 5)/( 3 n + 1 )

Simiplify : (16xx2^(n+1)-4xx2^(n))/(16xx2^(n+2)-2xx2^(n+1))

Observe the following pattern (1xx2)+(2xx3)=(2xx3xx4)/3, (1xx2)+(2xx3)+(3xx4)=(3xx4xx5)/3, (1xx2)+(2xx3)+(3xx4)+(4xx5)=(4xx5xx6)/3 and find the value of \ (1xx2)+(2xx3)+(3xx4)+(4xx5)+(5xx6)

Find the value of n : (2^(3)xx5^(n+1)xx10^(2)xx5^(n-1))/(125xx5^(n-2)xx2^(7))=(25)/(4)

Prove that 1^1xx2^2xx3^3xxxxn^nlt=[(2n+1)//3]^(n(n+1)//2),n in Ndot

Given that (1 + i)/(1 + 2^(2)i) xx (1 + 3^(2) i)/(1 + 4^(2) i) xx….xx (1 + (2n-1)^(2)i)/(1+(2n)^(2)i)= (a + bi)/(c+di) , show that (2)/(17) xx (82)/(257) xx ….xx ((2n-1)^(4) + 1)/((2n)^(4) + 1) = (a^(2) + b^(2))/(c^(2) + d^(2))

Simplify : (3xx27^(n+1)+9xx3^(3n-1))/(8xx3^(3n)-5xx27^(n))

Prove that : 1^3+2^3+3^3++n^3={(n(n+1))/2}^2dot