The nth term rule of the sequence 2, 6, 18, 54 is 2 * 3(n-1). This sequence is a geometric progression.
Understanding Geometric Progressions
A geometric progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant value called the common ratio.
Identifying the Pattern
In the sequence 2, 6, 18, 54:
- The first term (a) is 2.
- The common ratio (r) can be found by dividing any term by its preceding term. For example, 6 / 2 = 3, 18 / 6 = 3, and 54 / 18 = 3. Therefore, r = 3. As the reference states: to get the next term, multiply the one immediately preceding it by 3.
Deriving the nth Term Rule
The general formula for the nth term of a geometric progression is:
an = a * r(n-1)
Where:
- an is the nth term
- a is the first term
- r is the common ratio
- n is the term number
Substituting the values from our sequence:
an = 2 * 3(n-1)
Therefore, the nth term rule of the sequence 2, 6, 18, 54 is 2 * 3(n-1).
Examples
Term (n) | Calculation | Result |
---|---|---|
1 | 2 * 3(1-1) | 2 |
2 | 2 * 3(2-1) | 6 |
3 | 2 * 3(3-1) | 18 |
4 | 2 * 3(4-1) | 54 |