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Trying to prove that $1 + 2 +\dots + n = (n + 1)n/2$.
I have let $n = 1$ in the basis step, which lead to $1=1$, so it is true for $n = 1$.
For the induction step, I have assumed that $k\geq 1$ and let $n = k$ leading to:
$$1 + 2 + . . . + k = (k + 1)k/2$$
For which we then have to use to prove that the statement is valid for $n = k + 1$, which is where I get confused because this the statement is supposed to become:
$$1 + 2 + . . . + k + (k + 1) = (k + 2)(k + 1)/2$$
While I thought it would have just been
$$1 + 2 + . . . + (k + 1) = (k + 2)(k + 1)/2 $$
Via the substitution of $k = k + 1$. Why is there still a singular $k$ in the $k + 1$ statement?