Given
-2-8-32+...-32768
Find
Write the sum using the sigma notation
Explanation
we have given - 2 - 8 - 32 + ... - 32768
here a = -2
common ratio = -8/-2 = 4
nth term = -32768
now from the nth term formula we find the number of terms
[tex]\begin{gathered} a_n=ar^{n-1} \\ -32768=-2(4)^{n-1} \\ 16384=(4)^{n-1} \\ 4^7=(4)^{n-1} \\ 7=n-1 \\ n=8 \end{gathered}[/tex]
so , sum =
[tex]\sum_{i\mathop{=}1}^8a_ir^{i-1}=a_1r^{1-1}+a_2r^{2-1}+a_3r^{3-1}+.........+a_8r^{8-1}[/tex]
sum of geometric progression is given by
[tex]\begin{gathered} S_8=\frac{a(r^n-1)}{r-1} \\ \\ S_8=\frac{-2(4^8-1)}{8-1} \\ \\ S_8=\frac{-2(32768-1)}{7} \\ S_8=-43,690 \\ \end{gathered}[/tex]
Final Answer
[tex]\sum_{i\mathop{=}1}^8a_ir^{i-1}=-43,690[/tex]
