Master Arithmetic Sequences with Explicit and Recursive Rules

$n$	$a_{n}$	Rewrite
$1$	$a_{1}$	$a_{1} + 0 \cdot d$
$2$	$a_{2}$	$a_{1} + 1 \cdot d$
$3$	$a_{3}$	$a_{1} + 2 \cdot d$
$4$	$a_{4}$	$a_{1} + 3 \cdot d$
$5$	$a_{5}$	$a_{1} + 4 \cdot d$

n

a_{n}

Rewrite

1

a_{1}

a_{1} + 0 \cdot d

2

a_{2}

a_{1} + 1 \cdot d

3

a_{3}

a_{1} + 2 \cdot d

4

a_{4}

a_{1} + 3 \cdot d

5

a_{5}

a_{1} + 4 \cdot d

a Remember, it is already understood that

a_{1} = 50,

which means that the savings after the first week is

$ 50 .

Now, in order to find the savings for the second week, the recursive rule will be used for

n = 2 .

a_{n} = a_{n - 1} + 5

▼

Substitute values and evaluate

Substitute

$n = 2$

a_{2} = a_{2 - 1} + 5

SubTerm

Subtract term

a_{2} = a_{1} + 5

Substitute

$a_{1} = 50$

a_{2} = 50 + 5

AddTerms

Add terms

a_{2} = 55

The total savings after week

2

$ 55 .

The same procedure can be repeated for

n = 3

and

n = 4 .

It is important to follow this procedure in order, since any term value can only be figured out if the previous term is known — just like the value of

a_{1}

was needed to find the value of

a_{2} .

The following table shows a summary of the calculations.

$a_{1} = 50, a_{n} = a_{n - 1} + 5$
Position	Substitute	Simplify
$n = 1$	$a_{1} = 50$	$a_{1} = 50$
$n = 2$	$a_{2} = a_{1} + 5 ⇓ a_{2} = 50 + 5$	$a_{2} = 55$
$n = 3$	$a_{3} = a_{2} + 5 ⇓ a_{3} = 55 + 5$	$a_{3} = 60$
$n = 4$	$a_{4} = a_{3} + 5 ⇓ a_{4} = 60 + 5$	$a_{4} = 65$

b The given recursive rule can compared with the general form of a recursive rule. This helps to identify the first term

a_{1}

and the common difference

d,

which are needed to write the explicit rule.

General Recursive Rule a_{1}, a_{n} = a_{n - 1} + d Given Recursive Rule a_{1} = 50, a_{n} = a_{n - 1} + 5

It is now understood that

a_{1} = 50

and

d = 5 .

Next, use the general form of an explicit rule and substitute the previously identified values.

a_{n} = a_{1} + (n - 1) d ⇓ a_{n} = 50 + (n - 1) 5

Finally to find the value after a years worth of savings — 52 weeks — the explicit rule will be evaluated using

n = 52 .

a_{n} = 50 + (n - 1) 5

▼

Substitute

n

for

52

and evaluate

Substitute

$n = 52$

a_{52} = 50 + (52 - 1) 5

SubTerm

Subtract term

a_{52} = 50 + (51) 5

Multiply

a_{52} = 50 + 255

AddTerms

Add terms

a_{52} = 305

Davontay will save

$ 305

after one year.

c To help Davontay find the corresponding recursive rule, identify the first term

a_{1}

and the common difference

d

from the explicit rule given by the bank agent. This can be done by using the general form of an explicit rule.

General Explicit Rule a_{n} = a_{1} + (n - 1) d Given Explicit Rule a_{n} = 100 + (n - 1) 10

As seen,

a_{1} = 100

and

d = 10 .

Next, recall the general form of a recursive rule and substitute the values for

a_{1}

and

d .

a_{1}, a_{n} = a_{n - 1} + d ⇓ a_{1} = 100, a_{n} = a_{n - 1} + 10

Arithmetic Sequences

Catch-Up and Review

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Defining Sequences

A Special Type of Sequences: Arithmetic Sequences

Identifying Arithmetic Sequences

Explicit Rule of an Arithmetic Sequence

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Determining the Explicit Rule of an Arithmetic Sequence

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Describing Arithmetic Sequences Using a Recursive Rule

Recursive Rule of an Arithmetic Sequence

Writing a Recursive Rule for an Arithmetic Sequence

What to Do When No Consecutive Terms Are Known?

What to Do When $a_{1}$ Is Not Known?

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Answer

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Arithmetic Sequences

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	16 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Arithmetic Sequences

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Proof

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What to Do When No Consecutive Terms Are Known?

What to Do When a1​ Is Not Known?

Answer

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Answer

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Solution

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Arithmetic Sequences

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