SOLUTION: The third term of a geometric sequence is 12 and the fifth term is 163. Determine the formula for an, nth term of this sequence. Find the value of a⬇️25. (a25 with 25 on bo

Algebra ->  Test -> SOLUTION: The third term of a geometric sequence is 12 and the fifth term is 163. Determine the formula for an, nth term of this sequence. Find the value of a⬇️25. (a25 with 25 on bo      Log On


   

Question 1154626: The third term of a geometric sequence is 12 and
the fifth term is 163.
Determine the formula for an, nth term of this sequence.
Find the value of a⬇️25. (a25 with 25 on bottom)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
The third term of a geometric sequence is 12
a%5B3%5D=12
and the fifth term is 163
a%5B5%5D=163
the formula for an nth term of this sequence:
a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29

from given above we have
12=a%5B1%5D%2Ar%5E%283-1%29
12=a%5B1%5D%2Ar%5E2.......eq.1
and
163=a%5B1%5D%2Ar%5E%285-1%29
163=a%5B1%5D%2Ar%5E4.......eq.2

solve the system:
12=a%5B1%5D%2Ar%5E2.......eq.1
163=a%5B1%5D%2Ar%5E4.......eq.2
------------------------------------
12=a%5B1%5D%2Ar%5E2.......eq.1.....solve for a%5B1%5D
12%2Fr%5E2=a%5B1%5D.......eq.1a..........substitute in eq.2

163=%2812%2Fr%5E2%29%2Ar%5E4.......simplify
163=12r%5E2
163%2F12=r%5E2
r=sqrt%28163%2F12%29-> exact solution

go to eq.1a
12%2Fr%5E2=a%5B1%5D............substitute r
a%5B1%5D=12%2F%28sqrt%28163%2F12%29%29%5E2
a%5B1%5D=12%2F%28163%2F12%29
a%5B1%5D=144%2F163-> first term

then, the formula for an nth term of this sequence:

a%5Bn%5D=%28144%2F163%29%2A%28sqrt%28163%2F12%29%29%5E%28n-1%29

the value of a%5B25%5D will be

a%5B25%5D=%28144%2F163%29%2A%28sqrt%28163%2F12%29%29%5E%2825-1%29

a%5B25%5D=%28144%2F163%29%2A%28sqrt%28163%2F12%29%29%5E24

a%5B25%5D=2158060662623960090407387%2F61917364224

a%5B25%5D+=34853884522872%26+58007876059%2F61917364224-> exact solution

a%5B25%5D+ ≈34853884522873