Rewrite the sequence in the function form 2,6,18,54
The Correct Answer and Explanation is:
Function Form of the Sequence:
Given the sequence:
2, 6, 18, 54, …
This sequence follows a geometric pattern, where each term is obtained by multiplying the previous term by 3.
We can write the general term of this sequence using the formula for a geometric sequence:an=a⋅rn−1a_n = a \cdot r^{n-1}an=a⋅rn−1
Where:
- aaa is the first term
- rrr is the common ratio
- nnn is the term number
- ana_nan is the value of the nth term
For this sequence:
- a=2a = 2a=2 (the first term)
- r=3r = 3r=3 (since each term is 3 times the previous one)
Substitute these values into the formula:an=2⋅3n−1a_n = 2 \cdot 3^{n-1}an=2⋅3n−1
So, the function form of the sequence is:f(n)=2⋅3n−1f(n) = 2 \cdot 3^{n-1}f(n)=2⋅3n−1
Explanation
The sequence 2, 6, 18, 54 increases rapidly, suggesting a multiplicative pattern rather than an additive one. To determine the rule behind this sequence, observe how each term relates to the previous term:
- 6÷2=36 \div 2 = 36÷2=3
- 18÷6=318 \div 6 = 318÷6=3
- 54÷18=354 \div 18 = 354÷18=3
Since each term is three times the one before it, this sequence is geometric. A geometric sequence has a constant ratio between terms, and it can be described using a formula involving powers of the common ratio.
The general form of a geometric sequence is:an=a⋅rn−1a_n = a \cdot r^{n-1}an=a⋅rn−1
Here, aaa is the first term and rrr is the common ratio. For this particular sequence:
- The first term a=2a = 2a=2
- The common ratio r=3r = 3r=3
By substituting into the formula:an=2⋅3n−1a_n = 2 \cdot 3^{n-1}an=2⋅3n−1
This means that:
- a1=2⋅30=2a_1 = 2 \cdot 3^0 = 2a1=2⋅30=2
- a2=2⋅31=6a_2 = 2 \cdot 3^1 = 6a2=2⋅31=6
- a3=2⋅32=18a_3 = 2 \cdot 3^2 = 18a3=2⋅32=18
- a4=2⋅33=54a_4 = 2 \cdot 3^3 = 54a4=2⋅33=54
The formula accurately generates the terms of the sequence. This function form is useful when calculating any term in the sequence without listing all previous ones. For instance, the tenth term can be quickly found using:f(10)=2⋅39=39366f(10) = 2 \cdot 3^9 = 39366f(10)=2⋅39=39366
This shows how the formula simplifies work with large positions in the sequence.
