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.