Question

The sequence -2-11-26-47-74...-674 is neither arithmetic or geometric. How do I figure out the pattern to write it in summation notation?

Answer 1

first thing to do is looking at the difference between the terms.
It is here:
11-2 = 9
26-11 = 15
47-26= 21
74-47=27
...
You notice that the differences grow by a +6 term..
After 74 it should be the number N such that N-74 = 27+6 = 33.

Answer 2

that means, the sequence is
2 = 2
2+9 = 11
11+9+6 = 26
26+9+6+6 = 47
47+9+6+6+6 = 74

Answer 3

write out a little more, it might give something.

Answer 4

Trick: look at 74.
How many 9's in it?
How many 6's in it?
What about the next term? (107)

Answer 5

74 is the 5th term.
There are 5-1 "9"'s in the 5th term.
There are n-1 nines in the n-th term.

There are 1+2+3 6's in the 5th term.
There are 1+2+3+...+(n-2) sixes in the n-th term.

And there's 2.

So a_n = 2 + 9(n-1) + [1+2+...+(n-2)] * 6

Answer 6

And it seems to be correct.

Answer 7

by the way, you must use the formula for "1+2+...+k", with k=n-2 in order to obtain the real formula for \(a_n\).