To find the explicit rule for the nth term of the sequence 2, 8, 32, 128, we first need to analyze the relationship between consecutive terms.
The given sequence is:
- a1 = 2
- a2 = 8
- a3 = 32
- a4 = 128
Next, let’s examine how each term relates to the previous one:
- a2 = 2 × 4
- a3 = 8 × 4
- a4 = 32 × 4
We can see that each term is being multiplied by 4 to get the next term. To express this in another way, let’s see if we can write these terms in terms of powers of 4:
1. a1 = 2 = 2 × 40
2. a2 = 8 = 2 × 41
3. a3 = 32 = 2 × 42
4. a4 = 128 = 2 × 43
From these observations, we can deduce a pattern: the nth term can be expressed as:
an = 2 × 4(n-1)
To summarize, the explicit formula for the nth term of the given sequence is:
an = 2 × 4(n-1)
This formula allows you to calculate any term in the sequence by simply plugging in the value of n. For instance:
- If n = 1, a1 = 2 × 4(1-1) = 2 × 40 = 2
- If n = 2, a2 = 2 × 4(2-1) = 2 × 41 = 8
- If n = 3, a3 = 2 × 4(3-1) = 2 × 42 = 32
- If n = 4, a4 = 2 × 4(4-1) = 2 × 43 = 128
By utilizing this formula, you can not only identify terms in the sequence but also observe how the sequence grows exponentially as n increases.