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Summary IEB AP Math:Mathematical Induction
This document describes: -How to do both types of mathematical induction with workings (summing and dividing)
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Mathematical InductionLets say you had a friend. This friend wanted to prove a statement is true for a
certain condition.
For example:
Dividing
“I think that 5𝑛 − 1 is divisible by 4 for 𝑛 ∈ 𝑁
How would you go about proving it?
There is a step-by-step process to proving these type of statements:
STEP 1:
Prove the statement is true for 𝑛 = 1
5−1
=1
4
Therefore statement is true for 𝑛 = 1
We could now go 𝑛 = 2, 𝑛 = 3, 𝑛 = ⋯
But that would take forever, thus
STEP 2:
Let n=k and p= the output
This step is necessary to isolate one element, so we can prove 𝑛 = 𝑘 + 1. Like:
5𝑘 − 1
= 𝑝 𝑤ℎ𝑒𝑟𝑒 𝑝 ∈ 𝑁
4
5𝑘 = 4𝑝 + 1 𝑤ℎ𝑒𝑟𝑒 𝑝 ∈ 𝑁
In general: Isolate the element of the equation with 𝑏𝑎𝑠𝑒 𝑘
STEP 3:
Prove 𝑛 = 𝑘 + 1 is divisible by 4
Sub in 𝑛 = 𝑘 + 1 into 5𝑛 − 1
, 5𝑘+1 − 1
Now what?
Remember the exponent law:
𝑥 𝑎 × 𝑥 𝑏 = 𝑥 𝑎+𝑏
In this case:
5𝑘+1 = 5𝑘 × 5
But remember from step 2:
5𝑘 = 4𝑝 + 1
Thus:
5𝑘+1 − 1 = 5𝑘 × 5 − 1 = (4𝑝 + 1)(5) − 1
Now we must simply prove that this statement is divisible by 4.
Usually we aim for this type of format :
< 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 𝑒𝑞𝑎𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 > (𝑤ℎ𝑎𝑡𝑒𝑣𝑒𝑟)
So we are aiming for:
(4)(… )
Thus:
(4𝑝 + 1)(5) − 1
20𝑝 + 5 − 1
20𝑝 + 4
4(5𝑝 + 1)
Therefore based on the theory of mathematical induction, this statement is divisible
by 4 for n=1, n=2, n=3…
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