Write the recursive formula and the explicit formula for the sequence {3,6,12,24,48,...}. Show all work.
I think it is a1=3 an=an-1 times 2
To find the recursive formula and the explicit formula for the sequence {3,6,12,24,48,...}, we need to identify the pattern or relationship between consecutive terms.
If we observe the sequence carefully, we can see that each term is obtained by multiplying the previous term by 2. Mathematically, we can express this as:
aₙ = 2 * aₙ₋₁
where aₙ represents the nth term in the sequence and aₙ₋₁ represents the previous term.
Now, let's determine the initial term a₀. For this sequence, the first term is 3, so a₀ = 3.
Using the recursive formula, we can calculate the subsequent terms:
a₁ = 2 * a₀ = 2 * 3 = 6
a₂ = 2 * a₁ = 2 * 6 = 12
a₃ = 2 * a₂ = 2 * 12 = 24
a₄ = 2 * a₃ = 2 * 24 = 48
Thus, the recursive formula for this sequence is:
aₙ = 2 * aₙ₋₁, where a₀ = 3
To find the explicit formula, we need to determine the relationship between the term number (n) and the actual value of the term (aₙ). Let's examine the pattern:
a₀ = 3
a₁ = 6 = 2² * 3
a₂ = 12 = 2³ * 3
a₃ = 24 = 2⁴ * 3
a₄ = 48 = 2⁵ * 3
From the pattern, we can observe that each term is equal to 2 raised to the power of (n + 1) multiplied by the initial term a₀, which is 3 in this case. Mathematically, we can express the explicit formula as follows:
aₙ = 2ⁿ⁺¹ * a₀
Substituting a₀ = 3 into the explicit formula, we get:
aₙ = 2ⁿ⁺¹ * 3
Therefore, the explicit formula for the sequence is aₙ = 2ⁿ⁺¹ * 3.
To find the recursive formula and explicit formula for the sequence {3, 6, 12, 24, 48, ...}, we need to analyze the pattern in the given sequence.
First, let's observe the relationship between consecutive terms:
- The second term (6) is obtained by multiplying the first term (3) by 2.
- The third term (12) is obtained by multiplying the second term (6) by 2.
- The fourth term (24) is obtained by multiplying the third term (12) by 2.
- The fifth term (48) is obtained by multiplying the fourth term (24) by 2.
We can conclude that each term in the sequence is obtained by multiplying the previous term by 2.
Recursive formula:
Let's denote the nth term of the sequence as a[n].
From our observation, we can express the recursive formula as follows:
a[1] = 3 (the first term is 3)
a[n] = 2 * a[n-1] (for n > 1, each term is obtained by multiplying the previous term by 2)
Explicit formula:
The explicit formula provides a direct way to calculate any term based on its position in the sequence, without relying on previous terms. To find the explicit formula, we'll rewrite the recursive formula using index notation.
Using index notation, we can rewrite the recursive formula as:
a[1] = 3
a[n] = 2 * a[n-1] (for n > 1)
Now, let's manipulate the equation to isolate a[n]:
a[1] = 3
a[2] = 2 * a[1] = 2 * 3 = 6
a[3] = 2 * a[2] = 2 * 6 = 12
a[4] = 2 * a[3] = 2 * 12 = 24
We can observe that each term can be written as a power of 2:
a[n] = 2^(n-2) * a[2]
Thus, the explicit formula for the sequence is:
a[n] = 2^(n-2) * 3
To summarize:
- Recursive formula: a[n] = 2 * a[n-1] (for n > 1)
- Explicit formula: a[n] = 2^(n-2) * 3
clearly you just multiply by 2 each time, so it's a geometric sequence
a1 = 3
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