Well, let's line up the ones we do have, and see if
we can spot the pattern:
n=1 . . . . . 7
n=2 . . . . . 10
n=3 . . . . . 13
n=4 . . . . . 16
n=5 . . . . . 19
Notice that the next term is always 3 more than the one
where you are right now.
That means
(the change in the term) / (the change in 'n') = 3
That's the description of the 'slope' of a function.
Whatever the graph of terms vs. 'n' is, its slope is 3 .
The equation is something like T(n) = 3n .
Something like it, but not exactly.
If T(n) were exactly 3n, then T(1) ... the first term ... would be 3,
and it's not. It's 7 ... 4 more than that.
Similarly, if T(n) were exactly 3n, then T(2) ... the second term ...
would be 6, and it's not. It's 10 ... 4 more than that. Again !
Maybe that's it. Maybe T(n) = 3n + 4 .
Let's check it out.
If T(n) = 3n + 4, then T(3) ... the third term ... is (9 + 4) = 13. It is ! !
If T(n) = 3n + 4, then T(4) ... the fourth term ... is (12 + 4) = 16. It is ! !
Looking pretty good for that equation. Try the last one that's given:
If T(n) = 3n + 4, then T(5) ... the fifth term ... is (15 + 4) = 19. It is ! !
That must be it. For any number 'n', that term of
the series is
T(n) = 3n + 4 .
