Find the sum of the first five terms of the geometric series 8, −24, 72, … 484 488 648 684

Answer:The sum of first five terms are [tex]S_{5} = 8 -24 +72 -216 +648 = 488[/tex]Step-by-step explanation:Step 1:-sequence:-  an ordered pair of real numbers is called an sequenceExample:- { 1, 3, 5, 7, 9, ..........}and it is denoted by <[tex]a_{n}[/tex]series:- The sum of the sequence is called a series and it is denoted by [tex]S_{n}[/tex]The gives series is geometric series 8,-24,72,.......here a=8 and the ratio r=[tex]\frac{a_{2} }{a_{1} }[/tex] [tex]r= -3[/tex]Step 2:-Find The fourth term of the given sequenceGiven a=8 and r= -3[tex]t_{n}= a r^{n-1}[/tex][tex]t_{4} = 8(-3)^{4-1}[/tex][tex]t_{4}=8(-3)^3= -216[/tex]Find The fifth term of the given sequenceGiven a=8 and r= -3[tex]t_{n}=a r^{n-1}[/tex][tex]t_{5} = 8(-3)^{5-1}[/tex][tex]t_{5}=8(-3)^4= 648[/tex]Step 3:-now the geometric sequence  8,-24,72,-216,648sum of the geometric sequence is called geometric seriesThe first five terms of geometric series [tex]S_{5} } = 8 -24+72-216+648=488[/tex]orBy using sum of the Geometric series formula[tex]S_{n} =\frac{a(1-r^{n}) }{1-r}  if  r < 1[/tex]here a=8 and r = -3 <1[tex]S_{5} = \frac{8(1-(-3)^5}{1-(-3)} = 488[/tex]