This image shows a mathematical expression related to a step in a proof, likely part of an induction process for a series. The question asks which expression correctly represents the induction step for $S_k + 1$.

To analyze this:

1. The leftmost part contains sums of powers of integers up to $k$ (i.e., $1^2 + 2^2 + \cdots + k^2$).
2. The expression $\frac{1}{6}(k+1)(k+2)(2k+3)$ appears multiple times, which is typical for sums involving squares of integers.

We would need to identify which version correctly follows the pattern. Let's break it down:

• Expression A seems to maintain the correct form for the induction hypothesis.
• Expression B introduces an extra term or change that does not match the original progression.
• Expression C contains a misalignment in terms of the variables and coefficients.

Would you like to go deeper into any particular step of the process? I can guide you through a detailed explanation.

5 Follow-up questions:

1. How do we validate the base case in mathematical induction for this series?
2. What is the general formula for the sum of squares of the first $n$ natural numbers?
3. How does induction help to prove the sum formula for powers of integers?
4. What other methods exist for proving similar summations?
5. How can we further simplify the induction step for sums involving polynomial expressions?

Tip:

For clarity in mathematical induction, always verify both the base case and the inductive step before concluding that the proof holds.