<s>[INST] Adding $k^2$ to $1^2 + 2^2 + \cdots + (k - 1)^2$. In his book, Calculus Vol. 1, Tom Apostol mentions that adding $k^2$ to the predicate
$$A(k): 1^2 + 2^2 + \cdots + (k - 1)^2 < \frac{k^3}{3}$$
gives the inequality
$$ 1^2 + 2^2 + \cdots + k^2 < \frac{k^3}{3} + k^2.$$
Why does the RHS $$1^2 + 2^2 + \cdots + (k - 1)^2 $$ become $$ 1^2 + 2^2 + \cdots + k^2 $$ and not $$ 1^2 + 2^2 + \cdots + (k - 1)^2 + k^2$$
when adding $k^2$? 
Thank you.
 [/INST] The reason why,$$1^2 + 2^2 + \cdots + (k-1)^2,$$ becomes, $$1^2 + 2^2 + \cdots + (k - 1)^2+ k^2,$$ is because you're adding the $k$th term squared on both sides. Thus, instead of having the sum up to the $k-1$ th term squared, now you have the sum up to the $k$th term squared.
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