 # We want to prove the above statement by mathematical Induction for all natural numbers (n=1,2,3,…) Next SlideSlide 1.

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We want to prove the above statement by mathematical Induction for all natural numbers (n=1,2,3,…) Next SlideSlide 1

We need to show that for n=1 the statement is true. Next SlideSlide 2Previous Slide

Left hand side: Since when n=1, the first and last terms are the same, there is only 1 term on the left hand side. Right hand side: Plug in 1 for n. Next SlideSlide 3Previous Slide

Simplify the both sides to show that both sides are equal. Next SlideSlide 4Previous Slide

Since we have only proven the statement is true for n=1, we now are going to assume that it is true for an arbitrary natural number k. The word “Assume” is required in the proof to say that we are temporarily assuming this to be true even though we have not proven it yet. Next SlideSlide 5Previous Slide

As far as work for step #2, we just substitute k for n in the statement. Next SlideSlide 6Previous Slide

We now make use of our assumption in step #2 to prove for the next value of n, k+1. Next SlideSlide 7Previous Slide

Now we’ll substitute k+1 in for n. It’s a good idea to rewrite the statement showing the second to last term on the left hand side. Next SlideSlide 8Previous Slide

Simplify each side. Next SlideSlide 9Previous Slide

Note that part of the left hand side in step #3 is the same as the left hand side in step #2. Next SlideSlide 10Previous Slide

Now based on the equation in step #2 make a substitution. Next SlideSlide 11Previous Slide

We now have to show that both sides are the same. This can be done in many ways. I’ll start by adding the two 2 k terms. Just like x+x=2x, 2 k +2 k =2(2 k ). Next SlideSlide 12Previous Slide

Now I’ll Now I'll use properties of exponents and add the exponents of 2=2 1 and 2 k. Next SlideSlide 13Previous Slide

Finally left hand side is exactly the same as the right hand side. Next SlideSlide 14Previous Slide

We now know that if the statement is true for any natural number, k, it also has to be true for the next natural number, k+1. Thus, since we know it’s true for 1, it also has to be true for 2. Since it’s true for 2, it has to be true for 3. And so on, and so on. Therefore, we have proven the statement is true for all natural numbers, 1,2,3,…. Slide 15Previous SlideEND

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