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)?

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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)?

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Answer 1 · 2018-05-01T20:31:01

$S_{k} = \frac{k (k + 1) (k + 2)}{3}$ $S_k = (k(k+1)(k+2))/3$
$S_{k + 1} = \frac{(k + 1) (k + 2) (k + 3)}{3}$ $S_(k+1) = ((k+1)(k+2)(k+3))/3$

Explanation:

We are given that $S_{n} = \frac{n (n + 1) (n + 2)}{3}$ $S_n = (n(n+1)(n+2))/3$ .

To find $S_{k}$ $S_k$ , simply replace all occurrences of $n$ $n$ in $S_{n}$ $S_n$ with $k$ $k$ . That is, $S_{k} = \frac{k (k + 1) (k + 2)}{3}$ $S_k = (k(k+1)(k+2))/3$ .

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

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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)?

1 Answer

Explanation:

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