Write the explicit formula...
1. 0.75, 1.5, 3, 6, 12
2. 2, -3, -8, -13, -18
#1 doubles each time
#2 subtracts 5 each time
see what you can do with that
To find the explicit formula for a sequence, we need to determine the pattern in the sequence and use that to create a general formula.
1. For the sequence 0.75, 1.5, 3, 6, 12, we can observe that each term is obtained by multiplying the previous term by 2. Mathematically, this can be represented as:
a(n) = a(n-1) * 2
The first term, a(1), is 0.75.
Using this relationship, we can find the explicit formula for the sequence. Let's start with the second term:
a(2) = a(1) * 2 = 0.75 * 2 = 1.5
We can continue this process for the remaining terms:
a(3) = a(2) * 2 = 1.5 * 2 = 3
a(4) = a(3) * 2 = 3 * 2 = 6
a(5) = a(4) * 2 = 6 * 2 = 12
Therefore, the explicit formula for the sequence is:
a(n) = 0.75 * 2^(n-1)
2. For the sequence 2, -3, -8, -13, -18, we can observe that each term is obtained by subtracting 5 from the previous term. Mathematically, this can be represented as:
a(n) = a(n-1) - 5
The first term, a(1), is 2.
Using this relationship, we can find the explicit formula for the sequence. Let's start with the second term:
a(2) = a(1) - 5 = 2 - 5 = -3
We can continue this process for the remaining terms:
a(3) = a(2) - 5 = -3 - 5 = -8
a(4) = a(3) - 5 = -8 - 5 = -13
a(5) = a(4) - 5 = -13 - 5 = -18
Therefore, the explicit formula for the sequence is:
a(n) = 2 - 5(n-1)
To find the explicit formula for a sequence, we need to identify the pattern and the equation that generates each term in the sequence.
1. Looking at the first sequence, we can observe that each term is obtained by multiplying the previous term by 2. This means that each term is twice the previous term. To express this as an equation, we can denote the first term as a and write the equation as an = 2 * an-1, where n represents the position of the term in the sequence.
So, using this formula, we can find the explicit formula for the given sequence:
a1 = 0.75
a2 = 2 * a1 = 2 * 0.75 = 1.5
a3 = 2 * a2 = 2 * 1.5 = 3
a4 = 2 * a3 = 2 * 3 = 6
a5 = 2 * a4 = 2 * 6 = 12
Therefore, the explicit formula for the first sequence is an = 0.75 * 2^(n-1).
2. For the second sequence, we can observe that each term is obtained by subtracting 5 from the previous term. This means that each term is 5 less than the previous term. Using a similar approach, we can write the equation as an = an-1 - 5.
a1 = 2
a2 = a1 - 5 = 2 - 5 = -3
a3 = a2 - 5 = -3 - 5 = -8
a4 = a3 - 5 = -8 - 5 = -13
a5 = a4 - 5 = -13 - 5 = -18
Therefore, the explicit formula for the second sequence is an = 2 - 5(n-1).