Geometric Progression — Part 1: Common ratio and the nth term: visual lesson, ICSE Class 10 Mathematics

Lesson

Geometric Progression, Part 1

Multiply by the same number again and again, and the terms form a geometric progression.

In this lesson you will

Recognise a geometric progression and find its common ratio
Use the general term T_n = ar^{n−1} to find any term
Find which term has a given value, and find a and r from two given terms
Find a missing middle term and a term counted from the end

multiply by 2
add 3

Multiplying by the same number each time (a geometric progression) soon leaves adding behind.

Concept

What you will be able to do

Recognise a geometric progression and find its common ratio
Use the general term $T_n = ar^{n-1}$
Find which term has a given value; find $a$ and $r$ from two terms
Find a middle term and a term from the end

Concept

What is a geometric progression?

A list in which each term is the term before it multiplied by the same non-zero number $r$ , the common ratio: $r = \dfrac{T_{n+1}}{T_n}$ . The first term is called $a$ .

A geometric progression has the same ratio between every pair of neighbouring terms.
List	Each term ÷ the one before	Geometric progression?
$5, 10, 20, 40$	$2,\ 2,\ 2$	yes, $r = 2$
$1, -3, 9, -27$	$-3,\ -3,\ -3$	yes, $r = -3$
$16, 8, 4, 2$	$0.5,\ 0.5,\ 0.5$	yes, $r = \dfrac{1}{2}$
$2, 4, 8, 14$	$2,\ 2,\ 1.75$	no: the ratios differ

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