Exercises for Question Type 4: Using induction for proving equalities

Question 1

Skill question

Prove by induction that $1^{3} + 2^{3} + \dots + n^{3} = (\frac{n ( n + 1 )}{2})^{2}$ .

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Question 2

Skill question

Prove by induction that the sum of the first $n$ odd numbers is $1 + 3 + 5 + \dots + (2 n - 1) = n^{2}$

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Question 3

Skill question

Verify by induction that $1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}$ for all positive integers $n$ .

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Question 4

Skill question

Consider the sequence defined by $a_{1} = 2$ and $a_{n} = a_{n - 1} + 3 for n \geq 2 .$ Prove by induction that $a_{n} = 3 n - 1 for all n \geq 1 .$

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Question 5

Skill question

Prove by induction that $1^{2} + 2^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

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IB Mathematics Analysis and Approaches Question Type 4: Using Induction for Proving Equalities Exercises