(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that for every integer [tex]n\geq1[/tex], that

[tex]1^2 + 2^2 + ... + n^2 = \frac{n(n + 1)(2n + 1)}{6}[/tex]

2. The attempt at a solution

Note that everything after this sentence are what I am going to write in my assignment to hand in.

(1) Base Case

[tex]S(n) : \frac{n(n + 1)(2n + 1)}{6}[/tex]

Assume it is true for n = 1

[tex]S(1) : \frac{1(1 + 1)(2 + 1)}{6} = 1 = 1^2[/tex]

Which is true

(2) Inductive Step, let n = k + 1, then

[tex]S(k + 1) : \frac{(k + 1)(k + 2)(2k + 3)}{6}[/tex]

[tex]1^2 + 2^2 + ... + k^2 + (k + 1)^2 = \frac{k(k + 1)(2k + 1)}{6} + (k + 1)^2[/tex]

[tex]= \frac{k(k + 1)(2k + 1)}{6} + \frac{6(k + 1)^2}{6} = \frac{k(k + 1)(2k + 1) + 6(k + 1)^2}{6}[/tex]

[tex] = (k + 1)\frac{k(2k + 1) + 6(k + 1)}{6} = \frac{k + 1}{6} (2k^2 + 7k + 6) = \frac{k + 1}{6} (k + 2)(2k + 3) = \frac{(k + 1)(k + 2)(2k + 3)}{6}[/tex]

Therefore by Principles of Mathematical Induction, [tex]1^2 + 2^2 + ... + n^2 = \frac{n(n + 1)(2n + 1)}{6}[/tex] is true for all integers [tex]n\geq1[/tex]

Q.E.D

Questions

a) Is it okay to say "all" instead of "every"?

b) Yeah I will try to line it up in my real assignment, but I couldn't figure out how to do it here.

c) DO I have to put that Q.E.D thing to look smart?

d) Did I do it correctly or did I totally embarrassed myself

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# Homework Help: I can do algebra, but I can't prove this properly (and formally)

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