given that 2nd and the 3rd term of a geometric progression are 18 and 54 find the 5th term
ar = 18
ar^2 = 54
divide the 2nd by the 1st:
ar^2 / ar = 54/18
r = 3
then in ar = 18
3a = 18
a = 6
ar^4 = 6(3^4) = 486
or,
once we had r = 3
and knowing term3 = 54, then
term4 = 54*3 = 162
term5 = 162*3 = 486
To find the 5th term of the geometric progression, we need to find the common ratio (r) first.
The formula for a geometric progression is given by:
an = a1 * r^(n-1)
Where:
an = nth term of the geometric progression
a1 = first term of the geometric progression
r = common ratio of the geometric progression
n = term number
Given that the 2nd term (a2) is 18 and the 3rd term (a3) is 54, we can set up two equations to find the common ratio (r):
a2 = a1 * r
a3 = a1 * r^2
Substituting the given values:
18 = a1 * r ...(Equation 1)
54 = a1 * r^2 ...(Equation 2)
Dividing Equation 2 by Equation 1, we get:
54 / 18 = (a1 * r^2) / (a1 * r)
3 = r
Now that we have the common ratio (r = 3), we can find the 5th term (a5) using the formula:
a5 = a1 * r^(5-1)
Substituting the value of r = 3 and given that a2 = 18:
18 = a1 * 3
a1 = 18 / 3
a1 = 6
Using a1 = 6 and r = 3 in the formula for the 5th term:
a5 = 6 * 3^(5-1)
a5 = 6 * 3^4
a5 = 6 * 81
a5 = 486
Therefore, the 5th term of the geometric progression is 486.