The sum of a geometric sequence

Sequences and induction, grade 11 (5 units) Lesson 12 of 23Learning

The sum of a geometric sequence

The sum times

\displaystyle q

We write the sum, and under it the sum times $\displaystyle q$ – each term moves one place:

\begin{aligned}S_n&=a_1+a_1q+a_1q^2+\ldots+a_1q^{n-1}\\q\cdot S_n&=\phantom{a_1+{}}a_1q+a_1q^2+\ldots+a_1q^{n-1}+a_1q^n\end{aligned}

Subtracting, all the terms in the middle cancel: $\displaystyle {qS_n-S_n}={a_1q^n-a_1}$ .
$\displaystyle {S_n(q-1)}={a_1(q^n-1)}$ , so, when $\displaystyle q\neq1$ :

\displaystyle S_n

\displaystyle =

\displaystyle \frac{a_1(q^n-1)}{q-1}

\displaystyle q\neq1

When $\displaystyle {q}={1}$ all the terms are equal, and $\displaystyle {S_n}={n\cdot a_1}$ .

Example: the number of terms

We work out $\displaystyle 3+6+12+\ldots+768$ .

The number of terms: $\displaystyle {3\cdot2^{n-1}}={768}$ , that is $\displaystyle {2^{n-1}}={256}={2^8}$ , so $\displaystyle {n}={9}$ .
$\displaystyle {S_9}={\frac{3(2^9-1)}{2-1}}={3\cdot511}={1533}$ .

Alternating signs

The sum of the first 8 terms of $\displaystyle 2-6+18-\ldots$ ( $\displaystyle {a_1}={2}$ , $\displaystyle {q}={-3}$ ):

S_8=\frac{2\left((-3)^8-1\right)}{-3-1}=\frac{2\cdot6560}{-4}=-3280

The formula works when $\displaystyle q$ is negative too: it only needs brackets round $\displaystyle -3$ .

How many terms to pass a number

How many terms of $\displaystyle 1+2+4+\ldots$ must be added for the sum to pass 1,000?

$\displaystyle {S_n}={\frac{2^n-1}{2-1}}={2^n-1}$ .
By trial: $\displaystyle {2^9-1}={511}$ , $\displaystyle {2^{10}-1}={1023}$ .
10 terms are needed.

Geometric against arithmetic growth

An arithmetic sequence adds the same number at every step; a geometric sequence with $\displaystyle q>1$ multiplies by the same number at every step, so what it adds keeps growing.

$\displaystyle n$	1	5	10	15	20
$\displaystyle 100+50(n-1)$	100	300	550	800	1050
$\displaystyle 1\cdot2^{n-1}$	1	16	512	16384	524288

The geometric sequence starts far below the arithmetic one, and overtakes it by the eleventh term.

Exercise 1

Calculate the sum of the first 10 terms of $\displaystyle 5, 10, 20, \ldots$

/ – 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

The sum: $\displaystyle 1+3+9+\ldots+2187$ .

a

How many terms has the sum?

b

Calculate the sum.

/ – 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

Calculate the sum of the first 8 terms of $\displaystyle 64-32+16-\ldots$

/ – 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 4

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={3^n-1}$ .
Find $\displaystyle a_n$ as $\displaystyle S_n-S_{n-1}$ , and show that the sequence is geometric.

/ – 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 5

How many terms of $\displaystyle 2+6+18+\ldots$ must be added for the sum to pass 10,000? Find it by trial.

/ – 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

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

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