We can find the following pattern:
[tex]a_1 = 1 = (-1)^0\cdot\dfrac{1}{3^0}[/tex]
[tex]a_2 = -\dfrac{1}{3} = (-1)^1\cdot\dfrac{1}{3^1}[/tex]
[tex]a_3 = \dfrac{1}{9} = (-1)^2\cdot\dfrac{1}{3^2}[/tex]
[tex]a_4 = \dfrac{1}{27} = (-1)^3\cdot\dfrac{1}{3^3}[/tex]
We can deduce the generic rule
[tex]a_n = (-1)^{n-1}\cdot\dfrac{1}{3^{n-1}}[/tex]
And thus
[tex]a_{15} = (-1)^{14}\cdot\dfrac{1}{3^{14}}[/tex]
With this rule, you can compute the 5th, 6th, 7th, 8th, 9th and 10th element and sum them.
