Quick questions on Mathematical induction for series identities

The principle of mathematical induction lets you prove infinitely many statements at once from two finite checks. Let $P(n)$ be a statement about a positive integer $n$. If

The four-part structure is the template every series proof must follow, and you should write each part with its own heading or label so the marker can tick them off.

The four standard sum formulas come up constantly, both as the thing you must prove and as a result you can quote elsewhere:

The technique that drives the inductive step is always the same: peel off the final term so the rest matches the hypothesis. In symbols,

Verify $P(1)$ by substituting $n = 1$ into both sides separately and checking they are equal. Compute the two sides independently; never assume they match.

Assume $P(k)$ holds for some positive integer $k$, and write the assumed identity out in full. This line is the fact you are allowed to use later, so make it explicit.

Prove $P(k + 1)$ using the hypothesis. For a series, the move is mechanical: write the $(k + 1)$-term sum as the $k$-term sum plus the new last term, replace the $k$-term sum by the formula the hypothesis gives, then simplify to the closed form with $n = k + 1$ substituted.

State that since the base case holds and $P(k) \implies P(k + 1)$, by the principle of mathematical induction $P(n)$ holds for all positive integers $n$.

Factoring out $(k + 1)$ then matching the target form is the reliable route. If the algebra is messy, write the target form for $n = k + 1$ first and aim at it.

$\sum_{i = 1}^{n}$ runs from $i = 1$ to $i = n$. The $(k + 1)$-term sum includes the term at $i = k + 1$, which is exactly the new term you split off in the step.