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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1 Answer
Jarob R G.
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 = (n(6n^2 - 3n - 1))/2#. To check this, simply plug in the relevant value of #n#.

With #S_1#, we have #n=1#, giving #(1(6 - 3 - 1))/2 = 2/2 = 1#. This matches with what we have above.

For #S_2#, #n=2#, giving #(2(6*2^2 - 3*2 - 1))/2 = (2(24 - 6 - 1))/2 = 17#. This corresponds to what we found above.

Lastly, #S_3# has #n=3#, giving #(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# and #S_3# correspond to the formula for #S_n# provided.

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