Answer:
-21844
Step-by-step explanation:
Given:
- [tex]a_1=-4[/tex]
- [tex]a_n=-16384[/tex]
- [tex]r=4[/tex]
First find n by using the general form of a geometric sequence: [tex]a_n=ar^{n-1}[/tex] (where a is the first term and r is the common ratio)
[tex]\implies -16384=(-4)(4)^{n-1}[/tex]
[tex]\implies 4^{n-1}=\dfrac{-16384}{-4}[/tex]
[tex]\implies 4^{n-1}=4096[/tex]
[tex]\implies \ln 4^{n-1}=\ln 4096[/tex]
[tex]\implies (n-1)\ln 4=\ln 4096[/tex]
[tex]\implies n=\dfrac{\ln 4096}{\ln 4}+1[/tex]
[tex]\implies n=6+1[/tex]
[tex]\implies n=7[/tex]
Sum of the first n terms of a geometric series:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
(where a is the first term and r is the common ratio)
Substituting the given values and the found value of n into the formula:
[tex]\implies S_{7}=\dfrac{(-4)(1-4^7)}{1-4}[/tex]
[tex]\implies S_{7}=-21844[/tex]
