Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.

Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2

Answer 1

\[Sn: 1 + 4 + 7 + . . . + (3n - 2) = \frac{n(3n - 1)}{2}
\]
\[Sk: 1 + 4 + 7 + . . . + (3k - 2) = \frac{k(3k - 1)}{2}
\]

Answer 2

\[Sk: 1 + 4 + 7 + . . . + (3k - 2) = \frac{k(3k - 1)}{2}\]
\[Sk: 1 + 4 + 7 + . . . + (3(k+1) - 2) = \frac{(k+1)(3(k+1) - 1)}{2}\]

Answer 3

then algebra to "simplify" the last bit

Answer 4

for example
\[(3(k+1)-2)=3k+3-2=3k+1\]

Answer 5

and \[\frac{(k+1)(3(k+1) - 1)}{2}=\frac{(k+1)(3k+3) - 1)}{2}=\frac{(k+1)(3k+2)}{2}\]

Answer 6

\(S_{k+1}: 1 + 4 + 7 + . . . + (3(k+1) - 2) = \frac{(k+1)(3(k+1) - 1)}{2}\)?

Answer 7

I don't get it ? what are the dots ?