• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Exercise 2.3.9:

Why these ads ...

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. 2.0.1(b)
(c) 1994-2017, Bert G. Wachsmuth
Page last modified: Mar 3, 2017