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) ?

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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) ?

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Answer 1 · 2018-05-01T20:37:58

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

That is, $S_{1} = 1^{2} = 1$ $S_1 = 1^2 = 1$ . Also, $S_{2} = 1^{2} + 4^{2} = 17$ $S_2 = 1^2 + 4^2 = 17$ and $S_{3} = 1^{2} + 4^{2} + 7^{2} = 1 + 16 + 49 = 66$ $S_3 = 1^2 + 4^2 + 7^2 = 1 + 16 + 49 = 66$ .

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

With $S_{1}$ $S_1$ , we have $n = 1$ $n=1$ , giving $\frac{1 (6 - 3 - 1)}{2} = \frac{2}{2} = 1$ $(1(6 - 3 - 1))/2 = 2/2 = 1$ . This matches with what we have above.

For $S_{2}$ $S_2$ , $n = 2$ $n=2$ , giving $\frac{2 (6 \cdot 2^{2} - 3 \cdot 2 - 1)}{2} = \frac{2 (24 - 6 - 1)}{2} = 17$ $(2(6*2^2 - 3*2 - 1))/2 = (2(24 - 6 - 1))/2 = 17$ . This corresponds to what we found above.

Lastly, $S_{3}$ $S_3$ has $n = 3$ $n=3$ , giving $\frac{3 (6 \cdot 3^{2} - 3 \cdot 3 - 1)}{2} = \frac{3 (54 - 9 - 1)}{2} = \frac{3 (44)}{2} = 3 (22) = 66$ $(3(6*3^2 - 3*3 - 1))/2 = (3(54 - 9 - 1))/2 = (3(44))/2 = 3(22) = 66$ .

Thus, we've found that $S_{1}, S_{2}$ $S_1, S_2$ and $S_{3}$ $S_3$ correspond to the formula for $S_{n}$ $S_n$ provided.

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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

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