Question

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.(refer to image)?

Answer 1

`S_1`: `2 = 2`

`S_2`: `2+5 = 7`

`S_3`: `2+5+8 = 15`

Explanation:

The statement `S_n` claims that, if you sum the first `n` numbers of the form `3n-1`, the result will be `n*(3n+1)/2`

So, the statement `S_1` simply refers to the case `n=1`. This means that on the left hand side you don't really sum on anything, having one number alone. On the right hand side, you evaluate `n*(3n+1)/2`, given `n=1`. Let's compute both sides:

Since the generic number is expressed by `3n-1`, if `n=1` we have `3*1-1=2`. The left hand side is `2`.

The right hand side evaluates to `1*(3*1+1)/2 = 4/2 = 2`. The equation holds.

Let's deal with `S_2`, i.e. make the same considerations as above, but with `n=2`.

We have two sum the first two numbers: we already know that the first number is 2. The second number is `3*2-1 = 5`. So, the left hand side equals `2+5 = 7`.

The right hand side evaluates to `2*(3*2+1)/2 = 6+1 = 7`. The equation holds.

Similarly, when `n=3`, the third number is `3*3-1 = 8`. The left hand side is `2+5+8 = 15`

The right hand side evaluates to `3*(3*3+1)/2 = 3 * 10/2 = 3*5 = 15`. The equation holds.