# A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work. (refer to image)?

May 1, 2018

${S}_{k} = \frac{k \left(k + 1\right) \left(k + 2\right)}{3}$
${S}_{k + 1} = \frac{\left(k + 1\right) \left(k + 2\right) \left(k + 3\right)}{3}$

#### Explanation:

We are given that ${S}_{n} = \frac{n \left(n + 1\right) \left(n + 2\right)}{3}$.

To find ${S}_{k}$, simply replace all occurrences of $n$ in ${S}_{n}$ with $k$. That is, ${S}_{k} = \frac{k \left(k + 1\right) \left(k + 2\right)}{3}$.

To find ${S}_{k + 1}$, replace all instances of $n$ in ${S}_{n}$ with $k + 1$. This gives ${S}_{k + 1} = \frac{\left(k + 1\right) \left(\left(k + 1\right) + 1\right) \left(\left(k + 1\right) + 2\right)}{3}$. Combining terms, this gives ${S}_{k + 1} = \frac{\left(k + 1\right) \left(k + 2\right) \left(k + 3\right)}{3}$.