Exercises for Question Type 3: Verifying the proof

Question 1

Skill question

Verify that for $m = 3$ , the identity $1^{2} + 2^{2} + \dots + m^{2} = \frac{m ( m + 1 ) ( 2 m + 1 )}{6}$ holds.

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Question 2

Skill question

Verify that for $m = 3$ , the sum of cubes satisfies $1^{3} + 2^{3} + \dots + m^{3} = (\frac{m ( m + 1 )}{2})^{2}$ .

[3]

Question 3

Skill question

Verify that for $m = 3$ , the expression $m^{2} - m$ is even.

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Question 4

Skill question

Verify for $m = 3$ that the sum of the first $m$ odd numbers equals $m^{2}$ .

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Question 5

Skill question

Verify that for $n = 3$ , the inequality $n! \geq 2^{n - 1}$ holds.

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Question 6

Skill question

Expand and verify the binomial theorem for $n = 3$ in the expansion of $(a + b)^{n}$ .

[3]

Question 7

Skill question

Verify that for $m = 3$ , the geometric sum $2^{0} + 2^{1} + \dots + 2^{m} = 2^{m + 1} - 1$ holds.

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Question 8

Skill question

Verify for $m = 3$ that $k = 1 \prod m (1 + \frac{1}{k}) = m + 1$ .

[2]

Question 9

Skill question

Verify the base case $m = 3$ for the formula $1 + 2 + \dots + m = \frac{m ( m + 1 )}{2}$ .

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Question 10

Skill question

Verify that for $m = 3$ , $3$ divides $m^{3} - m$ .

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Question 11

Skill question

Verify for $n = 3$ that $5^{n} - 4 n - 1$ is divisible by $8$ .

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IB Mathematics Analysis and Approaches Question Type 3: Verifying the Proof Exercises