# 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)?

May 2, 2018

${S}_{1}$: $2 = 2$

${S}_{2}$: $2 + 5 = 7$

${S}_{3}$: $2 + 5 + 8 = 15$

#### Explanation:

The statement ${S}_{n}$ claims that, if you sum the first $n$ numbers of the form $3 n - 1$, the result will be $n \cdot \frac{3 n + 1}{2}$

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 $n \cdot \frac{3 n + 1}{2}$, given $n = 1$. Let's compute both sides:

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

The right hand side evaluates to $1 \cdot \frac{3 \cdot 1 + 1}{2} = \frac{4}{2} = 2$. 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 $3 \cdot 2 - 1 = 5$. So, the left hand side equals $2 + 5 = 7$.

The right hand side evaluates to $2 \cdot \frac{3 \cdot 2 + 1}{2} = 6 + 1 = 7$. The equation holds.

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

The right hand side evaluates to $3 \cdot \frac{3 \cdot 3 + 1}{2} = 3 \cdot \frac{10}{2} = 3 \cdot 5 = 15$. The equation holds.