Question

t1=40; tn=0.5tn-1 + n
find the explicit formula for the sequence from this recursion formula

Answer 1

I change a little bit about the notation to make it clear
T_1 = 40
T_n = 0.5 T( n-1) +n
---> \[T_n - 0.5T_{n-1}=n\]
RHS look for A such that \[T_n =An\]
\[0.5T_n= 0.5 A(n-1)\]
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\[T_n - 0.5T_{n-1} = A*n - 0.5 A(n-1)= n\]
---> 0.5A*n =n --> A = 2
so RHS = 2n for the required equation.
I stop here, If you know what I mean, I step up to calculate LHS and then combine them to give out the final result.
If you don't know what I am doing. I am sorry, cannot help