# 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)?

##### 1 Answer
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.