1. ## Arithmetic Series

I'm trying to find the number of terms in the arithmetic series $\displaystyle -4, 2, 8, 14, ....$ which makes the sum strictly greater than $\displaystyle 18,000$.

I know that $\displaystyle S_{n} = \frac{1}{2}\left[{2a+(n-1)d}\right]$, but I somehow can't apply it to this problem.

I'm trying to find the number of terms in the arithmetic series $\displaystyle -4, 2, 8, 14, ....$ which makes the sum strictly greater than $\displaystyle 18,000$.

I know that $\displaystyle S_{n} = \frac{1}{2}\left[{2a+(n-1)d}\right]$, but I somehow can't apply it to this problem.
You know a_1 and d. Set S_n=18,000 and solve for n.

3. Originally Posted by Jameson
You know a_1 and d. Set S_n=18,000 and solve for n.
$\displaystyle a = -4$

$\displaystyle d = 6$

$\displaystyle \frac{1}{2}\left[2(-4)+(n-1)6\right] = 18000$

$\displaystyle \frac{1}{2}\left[-8+6n-6\right] = 18000$

$\displaystyle \frac{1}{2}\left[-8+6n-6\right] = 18000$

$\displaystyle \frac{1}{2}\left[6n-14\right] = 18000$

3$\displaystyle n-7 = 18,000$

$\displaystyle 3n = 18,000 -7$

$\displaystyle 3n = 17993$

$\displaystyle n = \frac{17993}{3}$

$\displaystyle n = 5997.666667$

$\displaystyle 5998$ terms?

4. Almost. You subtracted 7 but should have added.

5. Originally Posted by Jameson
Almost. You subtracted 7 but should have added.
Oh, silly mistake.

$\displaystyle 3n = 18,000 +7$

$\displaystyle 3n = 18007$

$\displaystyle n = \frac{18007}{3}$

$\displaystyle n = 6002.333333$

$\displaystyle 6003$ terms then?

6. Seems to make sense.

7. Many thanks!