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Unformatted text preview: t see the PMI in many courses
yet to come. Sometimes it is explicitly stated and sometimes it remains hidden in the background.
If ever you see a property stated as being true ‘for all natural numbers n’, it’s a solid bet that the
formal proof requires the Principle of Mathematical Induction. 578 9.3.1 Sequences and the Binomial Theorem Exercises 1. Prove the following assertions using the Principle of Mathematical Induction.
n j2 = n(n + 1)(2n + 1)
6 j3 = (a) n2 (n + 1)2
4 j =1
(c) j =1
2n > (d) 3n 500n for n > 12 ≥ n3 for n ≥ 4 (e) Use the Product Rule for Absolute Value to show |xn | = |x|n for all real numbers x and
all natural numbers n ≥ 1
(f) Use the Product Rule for Logarithms to show log (xn ) = n log(x) for all real numbers
x > 0 and all natural numbers n ≥ 1.
for n ≥ 1.
2. Prove Equations 9.1 and 9.2 for the case of geometric sequences. That is:
(a) For the sequence a1 = a, an+1 = ran , n ≥ 1, prove an = arn−1 , n ≥ 1.
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