The ratio, the first term and the geometric mean

Sequences, grade 12 (4 units) Lesson 13 of 20Learning

The ratio, the first term and the geometric mean

Two terms given

In a geometric sequence $\displaystyle {a_2}={6}$ and $\displaystyle {a_5}={162}$ .
The solution below writes both terms with $\displaystyle a_1$ and $\displaystyle q$ , and divides one equation by the other:

\begin{cases}a_1q=6\\a_1q^4=162\end{cases}

Divide the second by the first: $\displaystyle {\frac{a_1q^4}{a_1q}}={\frac{162}{6}}$ , that is $\displaystyle {q^3}={27}$ , so $\displaystyle {q}={3}$ .
Substitute in the first: $\displaystyle {3a_1}={6}$ , so $\displaystyle {a_1}={2}$ .
The sequence: $\displaystyle 2, 6, 18, 54, 162, \ldots$

An even power: two sequences

In a geometric sequence $\displaystyle {a_3}={12}$ and $\displaystyle {a_5}={48}$ .
From $\displaystyle a_3$ to $\displaystyle a_5$ we multiply by $\displaystyle q$ twice: $\displaystyle {q^2}={\frac{48}{12}}={4}$ , so $\displaystyle {q}={2}$ or $\displaystyle {q}={-2}$ .

$\displaystyle {q}={2}$ : $\displaystyle {a_1}={\frac{12}{4}}={3}$ , and the sequence is $\displaystyle 3, 6, 12, 24, 48, \ldots$
$\displaystyle {q}={-2}$ : here too $\displaystyle {a_1}={3}$ , and the sequence is $\displaystyle 3, -6, 12, -24, 48, \ldots$

Both sequences fit the data.

The geometric mean

In a geometric sequence, the square of each term equals the product of the two terms on either side of it:

\displaystyle a_n^2

\displaystyle =

\displaystyle a_{n-1}\cdot a_{n+1}

\displaystyle n\geq2

The reason: $\displaystyle {a_{n-1}}={\frac{a_n}{q}}$ and $\displaystyle {a_{n+1}}={a_n\cdot q}$ , so their product is $\displaystyle a_n^2$ .
When the terms are positive, $\displaystyle {a_n}={\sqrt{a_{n-1}\cdot a_{n+1}}}$ – the geometric mean of its neighbours, hence the name “geometric sequence”. For example, in the sequence $\displaystyle 4, 8, 16$ : $\displaystyle {\sqrt{4\cdot16}}={8}$ .

Example: three neighbouring terms

The numbers $\displaystyle x-4$ , $\displaystyle x$ , $\displaystyle x+8$ (in this order) are three neighbouring terms of a geometric sequence.

$\displaystyle {x^2}={(x-4)(x+8)}={x^2+4x-32}$ .
$\displaystyle {4x-32}={0}$ , so $\displaystyle {x}={8}$ .
The numbers: $\displaystyle 4, 8, 16$ , and $\displaystyle {q}={2}$ .

The place of a term

In which place of the sequence $\displaystyle 3, 6, 12, \ldots$ is the term 96?

$\displaystyle {3\cdot2^{n-1}}={96}$ , so $\displaystyle {2^{n-1}}={32}$ .
$\displaystyle {32}={2^5}$ , so $\displaystyle {n-1}={5}$ , and $\displaystyle {n}={6}$ .

Both sides are written as powers of the same base, and the exponents are compared.

Exercise 1

In a geometric sequence $\displaystyle {a_2}={10}$ and $\displaystyle {a_5}={80}$ .

Find $\displaystyle q$ .

Find $\displaystyle a_1$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

In a geometric sequence $\displaystyle {a_3}={18}$ and $\displaystyle {a_5}={162}$ .

Find $\displaystyle q$ .

Write the first five terms of every sequence that fits the data.

Exercise 3

The numbers $\displaystyle x$ , $\displaystyle x+6$ , $\displaystyle x+24$ (in this order) are three neighbouring terms of a geometric sequence.
Find $\displaystyle x$ and the ratio.

\displaystyle q

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

The geometric sequence: $\displaystyle 3, 6, 12, \ldots$
In which place is the term 768?

Exercise 5

In a geometric sequence, the third term is 2 more than the second.
The fourth term is twice the third.
Find the first term.

Formulas: sequences, grade 12 (4 units)

Open as a page

A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$