If `t_n=(n+2)/((n+3)!)` then find the value of `sum_(n=5)^20 t_n`

If sum_(r=1)^(n)T_(r)=n(2n^(2)+9n+13); then find the value of sum_(r=1)^(n)sqrt(T_(r))

If sum_(r=1)^(n)T_(r)=(3^(n)-1), then find the sum of sum_(r=1)^(n)(1)/(T_(r))

The Fibonacci sequence is defence by t_(1)=t_(2)=1,t_(n)=t_(n-1)+t_(n-2)(n>2). If t_(n+1)=kt_(n) then find the values of k for n=1,2,3 and 4.

If sum_(r=1)^n T_r=(3^n-1), then find the sum of sum_(r=1)^n1/(T_r) .

If sum_(r=1)^n T_r=(3^n-1), then find the sum of sum_(r=1)^n1/(T_r) .

If sum_(r=1)^n T_r=(3^n-1), then find the sum of sum_(r=1)^n1/(T_r) .

Let T_(n) be the n^(th) term of a sequence for n=1,2,3,4... if 4T_(n+1)=T_(n) and T_(5)=(1)/(2560) then the value of sum_(n=1)^(oo)(T_(n)*T_(n+1)) is equal to

If t_(n)=an^(-1) , then find the value of n, given that a=2,r=3 and t_(n)=486 .