Answer :
1)
[tex]a_{1}=343\\ a_{r}=-1\\r=- \frac{1}{7} \\a_{r}=a_{1}* q^{r-1} \\-1=343*q^{r-1}\\-1=343* -(\frac{1}{7})^{r-1}\\-7^0=7^3*(- 7)^{-r+1}\\7^0=7^3*7^{-r+1}\\indexes: \\0=3-r+1\\n=4[/tex]
[tex] S_r=a_1 \frac{1-r^n}{1-r}\\S_r=343* \frac{1- (-\frac{1}{7} )^4}{1- (-\frac{1}{7} )}\\S_r=343* \frac{1- (\frac{1}{2401} )}{1+ (\frac{1}{7} )}\\S_r=343* \frac{(\frac{2400}{2401} )}{(\frac{8}{7} )}\\S_r=343* \frac{300}{343} \\S_r=300[/tex]
2)
[tex]S= a_{1}* \frac{1- r^{2} }{1-r}\\ S=80*\frac{1- ( -\frac{1}{2}) ^{2} }{1+\frac{1}{2} }\\S=80* \frac{1- \frac{1}{4} }{ \frac{3}{2} }=80* \frac{3}{4}* \frac{2}{3} =40[/tex]
3) we dont know how many numbers has this sequence
[tex]a_{1}=343\\ a_{r}=-1\\r=- \frac{1}{7} \\a_{r}=a_{1}* q^{r-1} \\-1=343*q^{r-1}\\-1=343* -(\frac{1}{7})^{r-1}\\-7^0=7^3*(- 7)^{-r+1}\\7^0=7^3*7^{-r+1}\\indexes: \\0=3-r+1\\n=4[/tex]
[tex] S_r=a_1 \frac{1-r^n}{1-r}\\S_r=343* \frac{1- (-\frac{1}{7} )^4}{1- (-\frac{1}{7} )}\\S_r=343* \frac{1- (\frac{1}{2401} )}{1+ (\frac{1}{7} )}\\S_r=343* \frac{(\frac{2400}{2401} )}{(\frac{8}{7} )}\\S_r=343* \frac{300}{343} \\S_r=300[/tex]
2)
[tex]S= a_{1}* \frac{1- r^{2} }{1-r}\\ S=80*\frac{1- ( -\frac{1}{2}) ^{2} }{1+\frac{1}{2} }\\S=80* \frac{1- \frac{1}{4} }{ \frac{3}{2} }=80* \frac{3}{4}* \frac{2}{3} =40[/tex]
3) we dont know how many numbers has this sequence
