The second-to-last and last lines of the proof refer to the inductive step in proving the formula for the sum of the first $n$ terms of an arithmetic sequence using mathematical induction. Let’s break down each line:

Second-to-Last Line (Inductive Hypothesis for $P_k$):

$S_{k+1} = S_k + u_{k+1}$ This is based on the fact that the sum of the first $k+1$ terms of an arithmetic sequence, $S_{k+1}$, is equal to the sum of the first $k$ terms, $S_k$, plus the next term in the sequence, $u_{k+1}$.

• $S_k$ is assumed to be true by the inductive hypothesis: $S_k = \frac{k}{2} \left( 2u_1 + (k-1)d \right)$ This equation comes from the known formula for the sum of the first $k$ terms of an arithmetic sequence.

• $u_{k+1}$, the $(k+1)$-th term of the sequence, is given by: $u_{k+1} = u_1 + kd$ This is the general form for the $(k+1)$-th term in an arithmetic sequence, where $u_1$ is the first term, and $d$ is the common difference.

Thus, combining the sum of the first $k$ terms and the $(k+1)$-th term gives the sum of the first $k+1$ terms.

Last Line (Simplification of $S_{k+1}$):

$S_{k+1} = \frac{k}{2} \left( 2u_1 + (k-1)d \right) + u_1 + kd$ In this line, we substitute the expression for $S_k$ from the inductive hypothesis into the equation for $S_{k+1}$.

Breaking it down:

1. $S_k$ is written as: $\frac{k}{2} \left( 2u_1 + (k-1)d \right)$
2. $u_{k+1}$ is written as $u_1 + kd$.

Now, we have to simplify this expression further. The goal is to show that this expression matches the desired form: $S_{k+1} = \frac{k+1}{2} \left( 2u_1 + k d \right)$ This will complete the inductive proof by showing that $P_{k+1}$ holds.

Would you like a detailed simplification of the final expression?

Here are 5 related questions for further exploration:

1. What is the significance of the base case in mathematical induction?
2. How is the general formula for the sum of an arithmetic series derived?
3. What role does the inductive hypothesis play in the proof process?
4. Can mathematical induction be used to prove geometric series sums?
5. How does the common difference $d$ affect the sum of an arithmetic sequence?

Tip: When proving by induction, always make sure your base case is thoroughly verified before moving to the inductive step.