Proof by Induction | Definition, Steps & Examples | Study.com - Free Printable

Let’s walk through this step by step — it’s a proof using something called “mathematical induction,” but don’t worry, we’ll keep it simple.

We’re trying to show that if the formula for adding numbers from 1 to k is:

> 1 + 2 + 3 + … + k = k(k+1)/2

…then the formula also works when we add one more number — (k+1) — so that we’re now adding up to (k+1):

> 1 + 2 + 3 + … + k + (k+1) = (k+1)(k+2)/2

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Step 1: Start with what we assume is true

We begin by assuming the formula works for k:

> 1 + 2 + 3 + … + k = k(k+1)/2

This is our starting point.

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Step 2: Add (k+1) to both sides

Since we want to find the sum up to (k+1), we just add (k+1) to both sides of the equation:

Left side becomes:
> 1 + 2 + 3 + … + k + (k+1)

Right side becomes:
> k(k+1)/2 + (k+1)

So now we have:

> 1 + 2 + 3 + … + k + (k+1) = k(k+1)/2 + (k+1)

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Step 3: Get a common denominator on the right side

The term (k+1) doesn’t have a denominator yet, but to combine it with k(k+1)/2, we write it as 2(k+1)/2:

> k(k+1)/2 + 2(k+1)/2

Now both terms have the same denominator.

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Step 4: Combine the numerators

Add the two fractions:

> [k(k+1) + 2(k+1)] / 2

Notice that both terms in the numerator have (k+1). That’s important!

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Step 5: Factor out (k+1)

Factor (k+1) from both parts of the numerator:

> (k+1)[k + 2] / 2

Which is the same as:

> (k+1)(k+2)/2

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Step 6: Conclusion

That’s exactly what we wanted to show! So if the formula works for k, then it also works for (k+1).

This completes the proof.

Final Answer:
(k+1)(k+2)/2