Question

Sigma notation help?

Answer 1

$\sum\limits_{n=1}^6\big[4(-5)^{n-1}\big]$

is a sum of the first six terms of a geometric series where first term:

$a_1=4\cdot(-5)^{1-1}=4\cdot(-5)^0=4\cdot1=\boxed{4}$

and common ratio $r = -5$. So the sum is:

$S_n=\dfrac{a_1(1-r^n)}{1-r}\\\\\\\\ S_6=\dfrac{4(1-(-5)^6)}{1-(-5)}=\dfrac{4(1-15625)}{6}=\dfrac{2\cdot(-15624)}{3}=\dfrac{-31248}{3}=\\\\\\=\boxed{-10416}$