A geometric progression has a common ratio of 3 if the difference between the 1st and 5th term is 160. Find the 5th term
To find the 5th term of a geometric progression with a common ratio of 3, we need to determine the value of the first term. Given that the difference between the 1st and 5th term is 160, we can use this information to calculate the value of the first term.
The formula to find the nth term of a geometric progression is:
an = a * r^(n - 1)
In this case, we have:
a1 = a
a5 = a * r^4
Given that the difference between a1 (the 1st term) and a5 (the 5th term) is 160, we can set up the following equation:
a5 - a1 = 160
a * r^4 - a = 160
Since the common ratio is 3, we can substitute r = 3:
a * (3^4 - 1) = 160
Now, we can simplify the equation:
a * (81 - 1) = 160
80a = 160
a = 160/80
a = 2
So, the first term (a1) is 2.
To find the 5th term (a5), we can use the formula again:
a5 = a * r^4
a5 = 2 * 3^4
a5 = 2 * 81
a5 = 162
Therefore, the 5th term of the geometric progression is 162.