Mastering Geometric Sequence Formula: Understanding Geometric Sequences

	$a_{1} > 0$	$a_{1} < 0$
$r > 1$	Increasing $3 \to \times 2 6 \to \times 2 12 \to \times 2 24 \to \times 2 48 \dots$	Decreasing $- 3 \to \times 2 - 6 \to \times 2 - 12 \to \times 2 - 24 \to \times 2 - 48 \dots$
$r = 1$	Constant $3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \dots$	Constant $- 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \dots$
$0 < r < 1$	Decreasing $48 \to \times \frac{1}{2} 24 \to \times \frac{1}{2} 12 \to \times \frac{1}{2} 6 \to \times \frac{1}{2} 3 \dots$	Increasing $- 48 \to \times \frac{1}{2} - 24 \to \times \frac{1}{2} - 12 \to \times \frac{1}{2} - 6 \to \times \frac{1}{2} - 3 \dots$
$r < 0$	Alternating $3 \to \times (- 2) - 6 \to \times (- 2) 12 \to \times (- 2) - 24 \to \times (- 2) 48 \dots$	Alternating $- 3 \to \times (- 2) 6 \to \times (- 2) - 12 \to \times (- 2) 24 \to \times (- 2) - 48 \dots$

a_{1} > 0

a_{1} < 0

r > 1

Increasing

3 \to \times 2 6 \to \times 2 12 \to \times 2 24 \to \times 2 48 \dots

Decreasing

- 3 \to \times 2 - 6 \to \times 2 - 12 \to \times 2 - 24 \to \times 2 - 48 \dots

r = 1

Constant

$3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \to \times 1 3 \dots$

Constant

$- 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \to \times 1 - 3 \dots$

0 < r < 1

Decreasing

48 \to \times \frac{1}{2} 24 \to \times \frac{1}{2} 12 \to \times \frac{1}{2} 6 \to \times \frac{1}{2} 3 \dots

Increasing

- 48 \to \times \frac{1}{2} - 24 \to \times \frac{1}{2} - 12 \to \times \frac{1}{2} - 6 \to \times \frac{1}{2} - 3 \dots

r < 0

Alternating

$3 \to \times (- 2) - 6 \to \times (- 2) 12 \to \times (- 2) - 24 \to \times (- 2) 48 \dots$

Alternating

$- 3 \to \times (- 2) 6 \to \times (- 2) - 12 \to \times (- 2) 24 \to \times (- 2) - 48 \dots$

$n$	$a_{n}$	Using $a_{1}$ and $r$
$1$	$a_{1}$	$a_{1} \cdot r^{0}$
$2$	$a_{2}$	$a_{1} \cdot r^{1}$
$3$	$a_{3}$	$a_{1} \cdot r^{2}$
$4$	$a_{4}$	$a_{1} \cdot r^{3}$

n

a_{n}

Using

a_{1}

and

r

1

a_{1}

a_{1} \cdot r^{0}

2

a_{2}

a_{1} \cdot r^{1}

3

a_{3}

a_{1} \cdot r^{2}

4

a_{4}

a_{1} \cdot r^{3}

a This situation can be modeled by a sequence with the first term

a_{1} = 1,

since a single grain is used on the first square, and a common ratio

r = 2,

since the terms double every time. First, the general form for the explicit rule of a geometric sequence will be recalled.

a_{n} = a_{1} \cdot r^{n - 1}

Now, the values

a_{1} = 1

and

r = 2

will be substituted to find the explicit rule of the presented sequence.

a_{n} = a_{1} \cdot r^{n - 1} ⇓ a_{n} = 1 \cdot 2^{n - 1}

b To find the total number of grains of rice of the king's debt, the explicit formula will be evaluated for

n = 64,

since the chessboard has

64

squares.

a_{n} = 1 \cdot 2^{n - 1}

▼

Substitute

64

for

n

and evaluate

Substitute

$n = 64$

a_{64} = 1 \cdot 2^{64 - 1}

SubTerm

Subtract term

a_{64} = 1 \cdot 2^{63}

CalcPow

Calculate power

a_{64} = 1 \cdot 9.223372 \dots \times 1 0^{18}

OneMult

$1 \cdot a = a$

a_{64} = 9.223372 \dots \times 1 0^{18}

RoundSigDig

Round to $3$ significant digit(s)

a_{64} = 9.22 \times 1 0^{18}

c First the total weight of the debt will be found. To do this, the number of grains will be multiplied by the weight of each individual grain:

4.4 \times 1 0^{- 6}

kilograms.

(9.22 \times 1 0^{18}) (4.4 \times 1 0^{- 6})

▼

Simplify

CommutativePropMult

Commutative Property of Multiplication

(9.22) (4.4) (1 0^{18}) (1 0^{- 6})

Multiply

40.568 (1 0^{18}) (1 0^{- 6})

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

40.568 (1 0^{12})

Write in scientific notation

4.0568 \times 1 0^{13}

That is

4.0568 \times 1 0^{13}

kilograms of rice. Recall that the whole Earth's production of rice per year is about

4.8 \times 1 0^{11}

kilograms. Therefore, the number of years needed for the whole planet to produce the rice of the debt can be obtained by dividing the total weight of the debt by the amount produced each year.

\frac{4 . 0 5 6 8 \times 1 0 ^{13}}{4 . 8 \times 1 0 ^{11}}

▼

Simplify

WriteProdFrac

Write as a product of fractions

\frac{4 . 0 5 6 8}{4 . 8} \cdot \frac{1 0 ^{13}}{1 0 ^{11}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

\frac{4 . 0 5 6 8}{4 . 8} \cdot 1 0^{2}

CalcPow

Calculate power

\frac{4 . 0 5}{4 . 8}

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Catch-Up and Review

How Does the Thickness of a Sheet of Paper Increase as It Is Folded in Half?

Predicting the Height of a Paper Sheet Folded Multiple Times

Introducing Geometric Sequences

Geometric Sequence

Identifying Geometric Sequences

Describing Geometric Sequences Using an Explicit Rule

Explicit Rule of Geometric Sequences

Proof

Modeling Multiplication of Bacteria Using Geometric Sequences

Answer

Hint

Solution

An Ecology Project that Follows a Geometric Sequence

Hint

Solution

The Geometric Sequence Behind the Story of Chess

Hint

Solution

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