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 1, 2018

The definition provided for ${S}_{n}$ provides you with the first three terms of ${S}_{n}$.

That is, ${S}_{1} = {1}^{2} = 1$. Also, ${S}_{2} = {1}^{2} + {4}^{2} = 17$ and ${S}_{3} = {1}^{2} + {4}^{2} + {7}^{2} = 1 + 16 + 49 = 66$.

We wish to check if each of these line up with the given formula of ${S}_{n} = \frac{n \left(6 {n}^{2} - 3 n - 1\right)}{2}$. To check this, simply plug in the relevant value of $n$.

With ${S}_{1}$, we have $n = 1$, giving $\frac{1 \left(6 - 3 - 1\right)}{2} = \frac{2}{2} = 1$. This matches with what we have above.

For ${S}_{2}$, $n = 2$, giving $\frac{2 \left(6 \cdot {2}^{2} - 3 \cdot 2 - 1\right)}{2} = \frac{2 \left(24 - 6 - 1\right)}{2} = 17$. This corresponds to what we found above.

Lastly, ${S}_{3}$ has $n = 3$, giving $\frac{3 \left(6 \cdot {3}^{2} - 3 \cdot 3 - 1\right)}{2} = \frac{3 \left(54 - 9 - 1\right)}{2} = \frac{3 \left(44\right)}{2} = 3 \left(22\right) = 66$.

Thus, we've found that ${S}_{1} , {S}_{2}$ and ${S}_{3}$ correspond to the formula for ${S}_{n}$ provided.