Geometric Sequences - A Guide

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Hi. In this math post I cover the topic of geometric sequences.


Pixabay Image Source

Screenshots are from witeboard.com with the use of my Wacom tablet & stylus.

Topics


  • Examples Of Geometric Sequences
  • Geometric Sequence Formula
  • Examples Of Solving For Missing Numbers In Geometric Sequences

 

Examples Of Geometric Sequences


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.

 

Geometric Sequence Formula


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.

geometricSequence_formula.PNG

 

Examples Of Using The Geometric Sequence Formula


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.

example01.PNG

 

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.

example02.PNG

 

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.

example03.PNG

 

Thank you for reading.

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When I looked into the applications of this, I was surprised it’s used in economics.

!discovery 31

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This geometric sequences topic is somewhat related to compound interest & time value of money.

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