Sequences
We know that a sequence is a group of numbers that are linked by a rule. That rule guide as so we can find the next numbers of the sequence. We want to find this sequence rule in order to find the 12th term.
In this case, the first term of the sequence is 7:
a₁ = 7
and the second is 21:
a₂ = 21
How are linked 7 and 21?
We can obtain 21 from 7 in two different ways:
adding 14
7 + 14 = 21
a₁ + 14 = a₂
multiplying by 3
3 · 7 = 21
3a₁ = a₂
These are the two possible rules applied in this sequence.
The third term of the sequence is 63:
a₃ = 63
How are linked the second term 21 and 63?
If we multiply 21 by 3 we obtain 63:
3 · 21 = 63
3 · a₂ = a₃
Remeber that 3a₁ = a₂, if we replace, we obtain:
3 · 3a₁ = a₃
3²a₁ = a₃
Then, the first three numbers of the sequence are linked by multiplying them by 3. The fourth term is obtained in the same manner:
3 · 63 = 189
3 · a₃ = a₄
Since 3²a₁ = a₃ then
3 · 3²a₁ = a₄
3³a₁ = a₄
Then the rule is "we obtain the next term of the sequence by multiplying the last number by 3"
Now, we can find some of the terms just by using this rule:
7, 21, 63, 189, 3 · 189 = 567, 3 · 567 = 1701, ...
{7, 21, 189, 567, 1701, ...}
Finding the 12th term is better if we express teh rule in mathematical terms:
Taking into account the expressions found previously, we can express this rule as:
aₙ = 3 · aₙ₋ ₁
aₙ = 3ⁿ⁻¹ · a₁
If we want to find the 12th term of the sequence we want to find a₁₂, this is n = 12. Using the last expression we have:
since 3¹¹ = 177,147 and a₁ = 7, then:
