In the problem below, It is asked to find the formula for the sum of the sequence and then to prove whether it is true or false for all n values using induction.
$$ 1 + 4 + 7 + ... + (3n + 1), \ n\in \Bbb N_0$$
In order to do that I tried to convert it into Sigma notation
$$\sum_{n=0}^k 3n + 1 $$
and then using the rules of sigma notation I came up with
$$\sum_{n=0}^k 3n + 1 = 3\cdot \sum_{n=0}^k n + \sum_{n=0}^k 1$$
and then I replaced it with the following to come to the formula for the sum of the sequence
$$3\cdot\frac{n(n+1)}{2} + (n + 1) = \frac{(n+1)(3n+2)}{2}$$
But it seems to be totally incorrect!
What am I doing wrong. Any help is appreciated.
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