The first four terms of a geometric progression are 2, 8, 32, 128
a) Write down the common ratio of the progression.
b) Write down the fifth term of the progression.
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)
