<span>3,−6,12,−24,...</span>
You can rewrite this as:
<span><span><span>(−1)</span>n</span>3⋅<span>(1,2,4,8,...)</span></span>
If we focus on <span>1,2,4,8...</span>
<span><span>a<span>n+1</span></span>=2<span>an</span></span>
with <span><span>a0</span>=1</span>.
This can be written out as a nice sum:
<span><span><span>N∑</span><span>n=0</span></span><span>2n</span>=<span>20</span>+<span>21</span>+<span>22</span>+<span>23</span>+...</span>
<span>=1+2+4+8+...</span>
Thus, now we can recombine everything to get:
<span>3<span><span>N∑</span><span>n=0</span></span><span><span>(−1)</span>n</span><span>2<span>n
</span></span></span>
