Where did this "+1" term come from for this inductive proof?

Question

Where did this "+1" term come from for this inductive proof? It is in boxed in black.

For context,

We are trying to prove this sequence:

has the following solution:

$$x_{ n }=\frac { 3^{ n+1 }-3-2n }{ 4 }$$

Answer 1

You wrote that $x_n$ is defined as $x_1=1$ and $x_k=3x_{k-1}+k$ for $k\geq2$. Hence the error is NOT the "$+1$" in the point you indicated: the mistake is when you write: $$ x_{k+1}=3x_k+k $$ in fact, according to definition, you here missed a "$+1$": $x_k=3x_{k-1}+k\Longrightarrow x_{k+1}=3x_k+(k+1)$. After this everything should be fixed.