In a GP if T3 = 18 and T6 = 486 Find:- T10

Accepted Solution

Answer:The 10th term of the G.P is 29.               Step-by-step explanation:Given : In a GP if [tex]T_3=18[/tex] and [tex]T_6 = 486[/tex].To find : The term [tex]T_{10}[/tex] ?Solution : The geometric sequence is in the form, [tex]a,ar,ar^2,ar^3,...[/tex]Where, a is the first term and r is the common ratio.The nth term of G.P is [tex]T_n=ar^{n-1}[/tex]We have given, [tex]T_3=18[/tex]i.e. [tex]T_3=ar^{3-1}[/tex][tex]18=ar^{2}[/tex] ....(1)[tex]T_6 = 486[/tex]i.e. [tex]a_6=ar^{6-1}[/tex][tex]486=ar^{5}[/tex] ....(2)Solving (1) and (2) by dividing them,[tex]\frac{486}{18}=\frac{ar^{5}}{ar^{2}}[/tex] [tex]27=r^3[/tex] [tex]r=\sqrt[3]{27}[/tex] [tex]r=3[/tex] Substitute in (1),[tex]18=a(3)^{2}[/tex] [tex]18=9a[/tex] [tex]a=2[/tex] The first term is a=2 and the common ratio is r=3.The 10th term, of GP is given by, [tex]T_{10}=2+(10-1)3[/tex][tex]T_{10}=2+(9)3[/tex][tex]T_{10}=2+27[/tex][tex]T_{10}=29[/tex]Therefore, The 10th term of the G.P is 29.