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An Example of Using Mathematical Induction

In this post will be given some examples as well as some banks about the problem of mathematical induction. You can learn from some examples and can then practice from some unanswered questions.

Example 1.

Prove that for all n\in\mathbb{N} then :

1+2+3+\cdots+n=\frac{1}{2}n\left(n+1\right)

Answer :

Step 1. (proving the statement is true for n=1)

For n=1

Then 1=\frac{1}{2}1\left(1+1\right) \longrightarrow (is true)

Step 2. (assume correct for n=k)

Assume correct for any n=k then it is true that

1+2+3+\cdots+k=\frac{1}{2}k\left(k+1\right)

Step 3.

(Prove true for n=k+1 with capital assumptions in step 2. So we have to prove that 1+2+3+\cdots+k+\left(k+1\right)=\frac{1}{2}\left(k+1\right)\left(k+2\right))

For n=k+1

Then

1+2+3+\cdots+k+\left(k+1\right)

    =\frac{1}{2}k\left(k+1\right)+\left(k+1\right)<span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://www.infinite-set.com/wp-content/ql-cache/quicklatex.com-024ed878b853acf437a4cf5cc647bdb3_l3.png" height="75" width="582" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[\longrightarrow$ we have capital assumptions that $1+2+3+\ldots+k=\frac{1}{2}k\left(k+1\right)$ $=\frac{k\left(k+1\right)+2\left(k+1\right)}{2}$ $=\frac{k^{2}+k+2k+2}{2}$ $=\frac{k^{2}+3k+2}{2}$ $=\frac{\left(k+1\right)\left(k+2\right)}{2}$ $=\frac{1}{2}\left(k+1\right)\left(k+2\right)\]" title="Rendered by QuickLaTeX.com"/>\blacksquare

I hope the explanation in the previous article and with the example above that I write a description in detail you can understand easily. So in the next example we will work with steps without much lengthy explanation because the basics are the same.

Example 2. (more…)

Mindset and Basic of Mathematical Induction

Mathematical induction is one of the tools of proof in mathematics. Usually mathematical induction is used to test the hypothesis. Some events make a pattern that eventually leads to a formula. But is this formula correct? Will this pattern apply to the next? Something like that.

Mindset of mathematical induction is like a domino game that is arranged upright then the first domino is dropped. Usually we call the domino effect. I think you already know about this game.

When the first dominoes fall will result in the second domino falling. And when the second domino falls then the third domino falls. And so on. This is if the domino order is correct. If something goes wrong, of course the domino effect will stop. That’s the mindset of mathematical induction.

Mathematic induction play on the proof that he fell on the first point, meanly that formula is true at first occurrence. Next, he assumes to fall on points to k. From that assumption, our task is to prove

that it falls on points to k+1. It’s mean that on points to k+1, he gets the domino effect from falling points to k. Already understand the mindset? We repeat in more detail :

Step 1. Prove he is right for the 1’st incident

Step 2. Assume he is right for the incident to k.

Step 3. Prove he is right for the incident to k+1 using assumed assumptions. (more…)