Certainly! Let's solve this step-by-step.
### a) Finding the Common Ratio
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the "common ratio."
Given the first four terms:
- Term 1 ([tex]\( a_1 \)[/tex]) = 2
- Term 2 ([tex]\( a_2 \)[/tex]) = 8
- Term 3 ([tex]\( a_3 \)[/tex]) = 32
- Term 4 ([tex]\( a_4 \)[/tex]) = 128
To find the common ratio ([tex]\( r \)[/tex]), we can use the following formula:
[tex]\[ r = \frac{a_{n+1}}{a_n} \][/tex]
where [tex]\( a_{n+1} \)[/tex] is the term immediately following [tex]\( a_n \)[/tex].
Let's find the common ratio step-by-step:
[tex]\[ r = \frac{a_2}{a_1} = \frac{8}{2} = 4 \][/tex]
To verify, let's check the other terms:
[tex]\[ r = \frac{a_3}{a_2} = \frac{32}{8} = 4 \][/tex]
[tex]\[ r = \frac{a_4}{a_3} = \frac{128}{32} = 4 \][/tex]
Hence, the common ratio [tex]\( r \)[/tex] is 4.
### b) Finding the Fifth Term
To find the fifth term, we use the formula for the [tex]\( n \)[/tex]-th term of a geometric progression:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Here, we need to find the 5th term ([tex]\( a_5 \)[/tex]):
- [tex]\( a_1 \)[/tex] = 2 (the first term)
- [tex]\( r \)[/tex] = 4 (the common ratio)
- [tex]\( n \)[/tex] = 5 (since we are finding the 5th term)
Substituting the values into the formula:
[tex]\[ a_5 = 2 \cdot 4^{(5-1)} \][/tex]
[tex]\[ a_5 = 2 \cdot 4^4 \][/tex]
[tex]\[ a_5 = 2 \cdot 256 \][/tex]
[tex]\[ a_5 = 512 \][/tex]
So, the fifth term of the geometric progression is 512.
### Summary:
a) The common ratio [tex]\( r \)[/tex] is 4.
b) The fifth term of the progression is 512.
