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I am currently studying proving by induction but I am faced with a problem.

I need to solve by induction the following question.

$$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$

for all $n > 1$.

Any help on how to solve this would be appreciated.

This is what I have done so far.

Show truth for $N = 1$

Left Hand Side = 1

Right Hand Side = $\frac{1}{2} (1) (1+1) = 1$

Suppose truth for $N = k$

$$1 + 2 + 3 + ... + k = \frac{1}{2} k(k+1)$$

Proof that the equation is true for $N = k + 1$

$$1 + 2 + 3 + ... + k + (k + 1)$$

Which is Equal To

$$\frac{1}{2} k (k + 1) + (k + 1)$$

This is where I'm stuck, I don't know what else to do. The answer should be:

$$\frac{1}{2} (k+1) (k+1+1)$$

Which is equal to:

$$\frac{1}{2} (k+1) (k+2)$$

Right?

By the way sorry about the formatting, I'm still new.

Martin Sleziak
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Andrew
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4 Answers4

5

Are you familiar with how induction works? We first prove this theorem for $n=1$. This is true because $$1 = \frac{1\cdot 2}{2}$$

Now assume that this is true for all values less than $n$, we try to show that it is true for $n$. We have $$1+2+\cdots+n-1+n = \frac{n\cdot (n-1)}{2}+n\\ =\frac{n\cdot (n-1)+2n}{2} = \frac{n^2+n}{2}= \frac{n(n+1)}{2}$$

So it's true for $n=1$, and if it's true for all values less than some number it's true for that number, which means it's true for all numbers.

guest196883
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  • This approach makes so much more sense to me using 1, 2, to n-1 to n. I can't follow how 1/2(k+1)(k+2) is equal to 1/2k(k+1) in the accepted answer... I guess bc the summation of k+1 would be equal to 1/2(k+1)(k+1+1) ? – nf071590 Sep 22 '23 at 19:02
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Basic algebra is what's causing the problems: you reached the point

$$\frac{1}{2}K\color{red}{(K+1)}+\color{red}{(K+1)}\;\;\;\:(**)$$

Now just factor out the red terms:

$$(**)\;\;\;=\color{red}{(K+1)}\left(\frac{1}{2}K+1\right)=\color{red}{(K+1)}\left(\frac{K+2}{2}\right)=\frac{1}{2}(K+1)(K+2)$$

DonAntonio
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0

Just pull out your induction-proof-template:

  • What is the base case ($n = 0$, presumably)? Does the formula work?
  • Now the induction step: If the formula is true for $n$, show that it is valid for $n + 1$. In this case (as in many sums) this is just to take the identity for $n$ and add the next term to both sides. The left hand side is already what you want, the right hand side probably needs some massage to put in the right form in terms of $n + 1$.

Details are left as exercise ;-)

vonbrand
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Think of pairing up the numbers in the series. The 1st and last (1 + n) the 2nd and the next to last (2 + (n - 1)) and think about what happens in the cases where n is odd and n is even.

If it's even you end up with n/2 pairs whose sum is (n + 1) (or 1/2 * n * (n +1) total)

If it's odd you end up with (n-1)/2 pairs whose sum is (n + 1) and one odd element equal to (n-1)/2 + 1 ( or 1/2 * (n - 1) * (n + 1) + (n - 1)/2 + 1 which comes out the same with a little algebra).

Lee Meador
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