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I)-indicates whether each progression is arithmetic, geometric, or neither. write a rule for each progression.1) 2,6,18,24,...2) 5, -10, 20, -40...3) 3,5,7,9,...4) 5,6,8,11,15...5) 1,2,6,24...6) 17,16,15,14...7) 50,-50,50,-50...8) 1,2,4,5,10,11,22...

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Progressions

Initial explanation

We know that a progression is a group of numbers that are linked by a rule.

That rule guide us so we can find the next numbers of the sequence.

We want to find the rule in these progressions.

- it is geometric if the rule shows a multiplication by a number.

- it is arithmetic if the rule shows an addition by a number.

- or neither if it is not any of both.

1). 2, 6, 18, 24, ...

In this case, the first term of the sequence is 2:

a₁ = 2

and the second is 6:

a₂ = 6

How are linked 2 and 6?

We can obtain 6 from 2 in two different ways:

adding 4

2 + 4 = 6

or multiplying it by 3

2 · 3 = 6

Since we can obtain the other numbers multiplying by 3:

a₁ = 2

a₂ = 2 · 3 = 6

a₃ = 6 · 3 = 18

a₄ = 18 · 3 = 54

Rule: multiplying by 3

Since the rule is a multiplication this progression is geometric.

2). 5, -10, 20, -40...

In this case, the first term of the sequence is 5:

a₁ = 5

and the second is -10:

a₂ = -10

We can obtain -10 from 5, multiplying 5 by -2 or substracting 15:

5 - 15 = -10

5 · (-2) = -10

Since we can obtain the other numbers multiplying by -2:

a₁ = 5

a₂ = 5 · (-2) = -10

a₃ = -10 · (-2) = 20

a₄ = 20 · (-2) = -40

Rule: multiplying by -2

Since the rule is a multiplication this progression is geometric.

3). 3, 5, 7, 9, ...

In this case, the first term of the sequence is 3:

a₁ = 3

and the second is 5:

a₂ = 5

We can obtain 5 from 3 adding 2:

3 + 2 = 5

Since we can obtain the other numbers adding 2:

a₁ = 3

a₂ = 3 + 2 = 5

a₃ = 5 + 2 = 7

a₄ = 7 + 2 = 9

Rule: adding 2

Since the rule is an addition, this progression is arithmetic.

4) 5, 6, 8, 11, 15...

In this case, we add 1, then 2, then 3, then 4:

a₁ = 5

a₂ = 5 + 1 = 6

a₃ = 6 + 2 = 8

a₄ = 8 + 3 = 11

Rule: adding 1, 2, 3

Since the rule is an addition of different numbers, this progression is not arithmetic nor geometric.

User Babatunde Adeyemi
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