The geometric mean, the first term and the ratio

Sequences and induction, grade 11 (5 units) Lesson 11 of 23Learning

The geometric mean, the first term and the ratio

The geometric mean

In a geometric sequence $\displaystyle {a_{n-1}}={\frac{a_n}{q}}$ and $\displaystyle {a_{n+1}}={a_n\cdot q}$ , so their product is $\displaystyle a_n^2$ :

\displaystyle a_n^2

\displaystyle =

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

\displaystyle n\geq2

When the terms are positive, $\displaystyle {a_n}={\sqrt{a_{n-1}\cdot a_{n+1}}}$ : the term is the geometric mean of its neighbours.
In the sequence $\displaystyle 4, 6, 9$ : $\displaystyle {6^2}={36}={4\cdot9}$ .

The ratio from two terms

The quotient of two terms gives a power of $\displaystyle q$ :

$\displaystyle {a_3}={12}$ and $\displaystyle {a_6}={96}$ : $\displaystyle {\frac{a_6}{a_3}}={q^3}={8}$ , so $\displaystyle {q}={2}$ and $\displaystyle {a_1}={\frac{12}{4}}={3}$ .
$\displaystyle {a_2}={8}$ and $\displaystyle {a_4}={32}$ : $\displaystyle {q^2}={4}$ , so $\displaystyle {q}={2}$ or $\displaystyle {q}={-2}$ – two sequences: $\displaystyle 4, 8, 16, 32, \ldots$ and $\displaystyle -4, 8, -16, 32, \ldots$
An even power of $\displaystyle q$ gives two solutions, and an odd power – one.

A system in

\displaystyle a_1

and

\displaystyle q

In a geometric sequence the sum of the first and the third term is 50, and the sum of the second and the fourth is 150.

$\displaystyle {a_1+a_1q^2}={50}$ and $\displaystyle {a_1q+a_1q^3}={150}$ .
The second equation is $\displaystyle {q\cdot(a_1+a_1q^2)}={150}$ , that is $\displaystyle {q\cdot50}={150}$ , so $\displaystyle {q}={3}$ .
$\displaystyle {a_1(1+9)}={50}$ , so $\displaystyle {a_1}={5}$ : the sequence $\displaystyle 5, 15, 45, 135, \ldots$

Three neighbouring terms

Three neighbouring terms of a geometric sequence are handy to write as $\displaystyle \frac{b}{q}$ , $\displaystyle b$ , $\displaystyle bq$ : in their product $\displaystyle q$ cancels.

The product of three neighbouring terms is 216, and their sum is 26.

$\displaystyle {\frac{b}{q}\cdot b\cdot bq}={b^3}={216}$ , so $\displaystyle {b}={6}$ .
$\displaystyle {\frac{6}{q}+6+6q}={26}$ , that is $\displaystyle {6q^2-20q+6}={0}$ , so $\displaystyle {q}={3}$ or $\displaystyle {q}={\frac{1}{3}}$ .
The numbers: $\displaystyle 2, 6, 18$ or $\displaystyle 18, 6, 2$ .

Exercise 1

In a geometric sequence $\displaystyle {a_2}={6}$ and $\displaystyle {a_5}={162}$ .
Find $\displaystyle a_1$ and $\displaystyle q$ .

Exercise 2

In a geometric sequence $\displaystyle {a_3}={18}$ and $\displaystyle {a_5}={162}$ .
Find $\displaystyle a_1$ and $\displaystyle q$ (two sequences).

Exercise 3

The numbers $\displaystyle x$ , $\displaystyle x+6$ , $\displaystyle 9x$ (in this order) are three neighbouring terms of a geometric sequence.
Find $\displaystyle x$ (two solutions), and the ratio for each.

Exercise 4

The product of three neighbouring terms of a geometric sequence is 1,000, and their sum is 35.
Call them $\displaystyle \frac{10}{q}$ , $\displaystyle 10$ , $\displaystyle 10q$ . Find $\displaystyle q$ and the numbers.

Exercise 5

In a geometric sequence the sum of the second and the third term is 36, and the sum of the third and the fourth is 108.
Find $\displaystyle a_1$ and $\displaystyle q$ .

Formulas: sequences and induction, grade 11 (5 units)

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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
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle S_n-S_{n-1}$ $\displaystyle n\geq2$
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
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n\left[2a_1+(n-1)d\right]}{2}$

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 a convergent sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$

The laws of powers

A product of powers of one base
$\displaystyle a^m\cdot a^n$ $\displaystyle =$ $\displaystyle a^{m+n}$
A quotient of powers of one base
$\displaystyle \frac{a^m}{a^n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^m\right)^n$ $\displaystyle =$ $\displaystyle a^{mn}$
A power with exponent 0
$\displaystyle a^0$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A power with a negative exponent
$\displaystyle a^{-n}$ $\displaystyle =$ $\displaystyle \frac{1}{a^n}$ $\displaystyle a\neq0$

Sums proved by induction

The sum of the natural numbers
$\displaystyle 1+2+3+\ldots+n$ $\displaystyle =$ $\displaystyle \frac{n(n+1)}{2}$
The sum of the odd numbers
$\displaystyle 1+3+5+\ldots+(2n-1)$ $\displaystyle =$ $\displaystyle n^2$
The sum of the squares
$\displaystyle 1^2+2^2+3^2+\ldots+n^2$ $\displaystyle =$ $\displaystyle \frac{n(n+1)(2n+1)}{6}$
The sum of the cubes
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$