Hi. In this math post I cover the topic of geometric sequences.
Screenshots are from witeboard.com with the use of my Wacom tablet & stylus.
To start here are a few examples of geometric sequences.
Example One
2, 4, 8, 16, 32, 64, 128
This sequence starts with 2 and doubles to obtain the next number until the last & seventh term of 128. Doubling is the same as multiplying by two. As the doubling is constant the common ratio here is 2.
Example Two - Divide By 10
90000, 9000, 900, 90, 9, 0.9, 0.09, 0.009
The first term here is ninety thousand. Obtaining the next number in the sequence involves dividing by 10. Dividing by ten is also the same as multiplying by one tenth. The common ratio here is 1/10 or 0.1. There are eight terms in this geometric sequence.
There is a formula for representing any number in a geometric sequence. In text form it is t_{n} = t1*(r)^(n-1). The n-th term is represented by t_{n}, the first term is represented by t_{1}, r is the common ratio and n is the number of terms. Screenshot shown below.
Example One
The common ratio in a geometric formula is 3 and the first term is 1/27. What is the fifteen term in this sequence?
With this question you have r = 3, t_{1} = 1/27 and n = 15. Use the geometric sequence formula to obtain the fifteen term.
Example Two
In this geometric sequence the starting number is 0.0001 with a common ratio of 10. How many terms are there from the starting number to one billion (1,000,000,000)?
The common ratio gives r = 10, t_{1} = 0.0001 and t_{n} = 1 000 000 000. Use the geometric sequence formula with the known values and solve for n. Note that one billion is ten to the power of nine.
Example Three
There are sixteen numbers from 1 to 32768 in a geometric sequence. What is the common ratio here?
The starting number is t_{1} = 1 and the last number is t_{15} = 32768.
Thank you for reading.
