Exercise 2.3.9:

Is the sum of the first n square numbers equal to (n + 2)/3 ?
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We might try to prove this statement via induction

Property Q(n):: 1² + 2² + 3² + ... + n² = (n + 2) / 3
Check Q(1):: 1² = 3/3 is true
Assume Q(n) is true:: Assume that 1² + 2² + 3² + ... + n² = (n + 2) / 3
Check Q(n+1):: 1² + 2² + 3² + ... + (n + 1)² =
= (1² + 2² + 3² + ... + n²) + (n+1)² =
= (n + 2) / 3 + 3 (n + 1)² / 3 =
= 1/3 (3 n² + 7n + 5)
which is not equal to (n+3)/3

Hence, the induction proof failed. That does not, in principle, mean that the statement is false. It is, however, a strong indication that property Q(n) is false. Indeed, checking for n = 2 gives:

Q(2) = 1 + 4 = 5 # 4 / 3

so that the statement is indeed false by this counterexample.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 26, 2007