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

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Expert Answers
ishpiro eNotes educator| Certified Educator

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.