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Question
Define the sequence a1, a2, a3 ... as follows:
a1 = 2, an = 5 an–1, for all natural numbers n ≥ 2.
Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula an = 2.5n–1 for all natural numbers.
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Solution
Let P(n) be the statement.
i.e., P(n): an = 2.5n–1 for all natural numbers,
We observe that P(1) is true,
Assume that P(n) is true for some natural number k
i.e., P(k): ak = 2.5k – 1.
Now to prove that P(k + 1) is true.
We have P(k + 1) : ak+1
= 5.ak
= 5.(2.5k – 1)
= 2.5k
= `2.5^((k + 1) - 1)`
Thus P(k + 1) is true whenever P(k) is true.
Hence, by the Principle of Mathematical Induction, P(n) is true for all natural numbers.
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