Need help with this question - Thanks: Use mathematical induction to prove the formula on the image below.

1 Answer | Add Yours

ishpiro's profile pic

ishpiro | College Teacher | (Level 1) Educator

Posted on

The method of mathematical induction consists of two steps:

1) Show that the formula is true for n = 1.

2) Show that if the formula is true for n = k, then it is also true for n = k+1.

For the given formula, it is easy to check that it holds for n = 1:

`3 = (3*1*(1+1))/2 = 3`

For step 2, assume that the formula is true for n = k, that is

`3 + 6 + 9 + ... + 3k = (3k(k+1))/2`

Now we need to show that the formula is then also true for n = k+ 1, which means that

`3 + 6 + 9 + ... + 3k + 3(k+1) = (3(k+1)(k+1+1))/2=(3(k+1)(k+2))/2`

Since we have assumed that the formula for n = k is true, we can add 3(k+1) to the left side of the formula, which will make it equal the left side of the formula for n = k+1. Then, we will have to show that the right sides are also equal, that is, that

`(3k(k+1))/2 + (3(k+1)) = (3(k+1)(k+2))/2`

Let's simplify the expression on the left side to show that it equals the expression on the right side. Start with factoring out 3(k+1):

`(3k(k+1))/2 + 3(k+1) = 3(k+1)(k/2 + 1) = `

`=3(k+1)(k+2)/2 = (3(k+1)(k+2))/2` .

We can see that the two expressions are in fact identical, so if the formula is true for n = k, it follows that it is true for n = k+1. This satisfies the second step of the method of induction.

We have shown that the formula is true according to the method of mathematical induction.

We’ve answered 318,929 questions. We can answer yours, too.

Ask a question