The infinite geometric sequence(Xn)?
1 Answer
Explanation:
#(a)#
#"the fist term in the sequence is "a=6.5#
#"the common ratio r is found as follows"#
#r=a_2/a_1=a_3/a_2= ...... =a_n/a_(n-1)#
#rArrr=(-27.3)/6.5=114.66/(-27.3)=-4.2#
#"sequence "=a,ar,ar^2,ar^3, ...... ,ar^(n-1)#
#(b)#
#"the n th term is"#
#a_n=ar^(n-1)#
#(c)#
#"here "a=6.5,r=-4.2" and "n=8#
#rArra_8=6.5xx(-4.2)^7~~-149850.567" 3 dec. places"#
#(d)#
#"the sum to n terms of a geometric sequence is"#
#•color(white)(x)S_n=(a(1-r^n))/(1-r)=(a(r^n-1))/(r-1)#
#"if "-1 < r<1to|r|<1#
#"then the sum will converge on a particular value"#
#"as "ntooo,r^nto0#
#rArrS_n=(a(1-0))/(1-r)=a/(1-r)larrcolor(blue)"sum to infinity"#
#"and is written as "S_oo=a/(1-r)#
#"if "r>1," then as "ntooo,r^ntooo" and "S_nto+-oo#
#"dependent on the sign of a"#
#"if "r<-1," then as "ntooo#
#"here " r=-4.2<-1#
#"as an example"#
#S_(30)=(6.5((-4.2)^(30)-1))/(-4.2-1)larrcolor(blue)"large and negative"#
#S_(35)=(6.5((-4.2)^(35)-1))/(-4.2-1)larrcolor(blue)"large and positive"#
