Sum disturbance method, find a compact form of the sum

Sum disturbance method, find a compact form of the sum

Postby Guest » Sun May 31, 2020 11:08 am

I tried to solve this example, but without success, could someone help me solve it because I got stuck on it and don't understand how to solve it.
Using the sum disturbance method, find a compact form of the following sum:

[tex]\textrm{(a)} \sum_{k=1}^n{(1+k2^{k-1})^2}[/tex]

Spoiler: show
The disturbance method is sum
[tex]{s_{n+1} = a_{1} + a_2 + \dots + a_n + a_{n+1}}[/tex]
expressed in two ways. The first is obvious:
[tex]s_{n+1} = s_n + a_{n+1}.[/tex]
The second is to present [tex]s_{n+1}[/tex] in form:
[tex]s_{n+1} = a_1 + f(s_n),[/tex]
where f is a function. Then we get the equality
[tex]s_n + a_{n+1} = a_1 + f(s_n).[/tex]
This equality can be treated as an equation with one unknown sn. When it is solved in relation to this unknown, we obtain a compact form of a sum.
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