March 06, 2020
You don’t need to be a math whiz to be a good programmer, but there are a handful of tricks you will want to add to your problem solving bag to improve the performance of your algorithms and make an impression in technical interviews. In this tutorial, you will learn proof by induction by summing consecutive integers 1 to n.
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Proof by induction is a mathematical method used to prove that a statement is true for all natural numbers. It’s not enough to prove that a statement is true in one or more specific cases. We need to prove it is true for all cases.
There are two metaphors commonly used to describe proof by induction:
Given a chain of dominos, if one falls, they will all fall.
Given a sturdy ladder, if one rung holds, they will all hold as we climb higher.
Let’s get philosophical.
Inductive reasoning and deductive reasoning are two methods of reason in logic.
We can think of them as opposites.
Deductive reasoning is top-down.
We start with general properties that are true and from them determine the truth of a specific property.
The classic example is a syllogism:
Inductive reasoning is bottom-up.
We start with specific properties, look for patterns, and make generalizations.
That’s not good enough!
We skeptics need proof.
A mathematical proof shows that “the stated assumptions logically guarantee the conclusion”.
All the time.
Despite its name, proof by induction is a method of deduction.
We need to prove that the specific case is true for all cases, without exception. Inductive reasoning doesn’t guarantee this.
Unlike deductive reasoning, we don’t start with a general rule.
We start with the rule we want to prove and assume it is true and then use mathematics to prove it generally.
There are two steps to proof by induction:
We first need to prove that our property holds for a natural number.
That’s generally 0 or 1.
What’s a ‘natural number’?
Natural numbers are the numbers we use for counting or ordering.
Once we establish a base case, we need to prove that property holds for the next natural number.
What’s the next natural number?
n + 1
Have we seen this, or something like it, before?
We can use proof by induction to prove the following:
1 + 2 + 3 + … + n = n * (n + 1) / 2
If this is new to you, you may want to start with How to Sum Consecutive Integers from 1 to n.
Let’s plug in values.
Our equation is
n * (n + 1) / 2
If n = 1, the result is 1.
If n = 2, the result is 3.
If n = 3, the result is 6.
This is the ‘brute force’ approach.
Following this approach, the only way to prove our equation works for all natural numbers is to calculate it for all natural numbers.
There must be a better way!
Let’s refer to our equation with a variable so I can type less.
P = n * (n + 1) / 2
Now we can simply refer to it as
We proved that our equation works when
n = 1.
What if we don’t know
Let’s add another variable to the equation,
Let’s say that
k is less than or equal to
k <= n
We need to make a proposition. What is it?
P(k) is true, then
P(k + 1) is also true.
If that’s true, then P(n) is true for all natural numbers.
Let’s rewrite our equation with
1 + 2 + 3 + … + k = k * (k + 1) / 2
That looks familiar. But we need to prove that
k + 1 works. If the number following
k + 1, we can add
k + 1 to the left of our equation:
1 + 2 + 3 + … + k + (k + 1) = k * (k + 1) / 2
If we add
k + 1 to the left of our equation, we also need to add it to the right:
1 + 2 + 3 + … + k + (k + 1) = k * (k + 1) / 2 + (k + 1)
Now we need to simplify.
What’s our common denominator?
We multiply our newly added
(k + 1) by 2.
1 + 2 + 3 + … + k + (k + 1) = k * (k + 1) / 2 + 2 * (k + 1) / 2
Now we can add the terms:
1 + 2 + 3 + … + k + (k + 1) = k * (k + 1) + 2 * (k + 1) / 2
Do we see a pattern? There are two
(k + 1) terms, so let’s factor them out:
1 + 2 + 3 + … + k + (k + 1) = (k + 1) * (k + 2) / 2
This is starting to look familiar. What is another way we can describe
(k + 2)?
((k + 1) + 1)
1 + 2 + 3 + … + k + (k + 1) = (k + 1) * ((k + 1) + 1) / 2
Did you catch that?
We just proved our equation!
1 + 2 + 3 + … + n = n * (n + 1) / 2
In this tutorial, you learned proof by induction by proving an equation to sum consecutive integers from 1 to n. Proof by induction is useful for understanding and calculating the Big O of recursive algorithms. We’ll look at that in a future article. Stay tuned!
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