Operations on a geometric sequence and sub-sequences

Sequences and induction, grade 11 (5 units) Lesson 13 of 23Learning

Operations on a geometric sequence and sub-sequences

Operations that keep a sequence geometric

$\displaystyle a_n$ is a geometric sequence with ratio $\displaystyle q$ . These sequences are geometric too:

$\displaystyle k\cdot a_n$ ( $\displaystyle k\neq0$ ): the ratio stays $\displaystyle q$ .
$\displaystyle a_n^2$ : $\displaystyle {\frac{a_{n+1}^2}{a_n^2}}={q^2}$ .
$\displaystyle \frac{1}{a_n}$ : the ratio is $\displaystyle \frac{1}{q}$ .
$\displaystyle a_{2n}$ , the terms in the even places: between them are two steps, and the ratio is $\displaystyle q^2$ .
$\displaystyle a_n\cdot b_n$ , where $\displaystyle b_n$ is a geometric sequence with ratio $\displaystyle p$ : the ratio is $\displaystyle q\cdot p$ .

The proof in every case: the quotient of two neighbouring terms of the new sequence is the same number for every $\displaystyle n$ .

An operation that breaks it

Adding a number to every term breaks a geometric sequence: $\displaystyle 1, 2, 4$ gives $\displaystyle 2, 3, 5$ , and the quotients $\displaystyle \frac{3}{2}$ and $\displaystyle \frac{5}{3}$ are not equal.

From an arithmetic sequence to a geometric one

If $\displaystyle a_n$ is an arithmetic sequence with difference $\displaystyle d$ , then $\displaystyle {b_n}={2^{a_n}}$ is a geometric sequence:

\frac{b_{n+1}}{b_n}=\frac{2^{a_{n+1}}}{2^{a_n}}=2^{a_{n+1}-a_n}=2^d

The constant difference in the exponent becomes a constant ratio.
$\displaystyle 1, 3, 5, 7, \ldots$ gives $\displaystyle 2, 8, 32, 128, \ldots$ , with ratio $\displaystyle {2^2}={4}$ .

A sequence of two geometric sequences

The sequence $\displaystyle 1, 2, 3, 4, 9, 8, 27, 16, \ldots$ interleaves two geometric sequences:

in the odd places $\displaystyle 1, 3, 9, 27, \ldots$ – ratio $\displaystyle 3$ ;
in the even places $\displaystyle 2, 4, 8, 16, \ldots$ – ratio $\displaystyle 2$ .

The sum of the first 12 terms: 6 terms of each sequence,

\frac{3^6-1}{3-1}+\frac{2(2^6-1)}{2-1}=364+126=490

Exercise 1

$\displaystyle a_n$ is a geometric sequence with ratio $\displaystyle {q}={-2}$ .
Which of these sequences are geometric? For each geometric one write its ratio, and for each other one – a counterexample.

$\displaystyle 5a_n$
$\displaystyle a_n^2$
$\displaystyle a_n+1$
$\displaystyle a_{3n}$
$\displaystyle \frac{1}{a_n}$

Exercise 2

$\displaystyle {a_n}={a_1q^{n-1}}$ is a geometric sequence, and $\displaystyle {b_n}={a_n\cdot a_{n+1}}$ .
Simplify $\displaystyle \frac{b_{n+1}}{b_n}$ , and explain what it proves.

/ – 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 3

The sequence: $\displaystyle 2, 1, 6, 4, 18, 16, \ldots$ (one geometric sequence in the odd places, and another in the even places).

Calculate $\displaystyle a_{11}$ .

/ – 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

Calculate $\displaystyle a_{12}$ .

/ – 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

Calculate the sum of the first 12 terms.

/ – 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

In a geometric sequence of 8 terms, the sum of the terms in the even places is 3 times the sum of the terms in the odd places.

Explain why $\displaystyle {q}={3}$ . (A hint: $\displaystyle {a_2}={a_1q}$ , $\displaystyle {a_4}={a_3q}$ , …)

The sum of all 8 terms is 6,560. Find $\displaystyle a_1$ .

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
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$