find the explicit formula.
1. 0.5, -1, 2, -4 ...
2. 5, -5/3, 5/9, -5/27 ...
3. 1.25, 5, 20, 80 ....
4. -4, -24, -144, -864 ...
5. 4, 12, 36, 108 ...
6. 3, -15, 75, -375 ...
7. 2, 4, 8, 16 ...
8. 3/2, 3/4, 3/8, 3/16 ...
9. -1, -6, -36, -216 ...
10. 3, 12, 48, 192 ...
1. geometric series
a r^(n-1)
a = .5, r = -2
.5 *(-2)^(n-1)
2. same deal
5 *(-1/3)^n-1
and on and on .....
To find the explicit formula of a sequence, we need to identify the pattern in the sequence and determine the rule that generates each term.
Let's analyze each sequence one by one:
1. 0.5, -1, 2, -4 ...
The pattern in this sequence is that each term alternates between multiplying the previous term by -2 and dividing the previous term by 2. So, the explicit formula for this sequence is: a(n) = (-2)^(n-1) * 0.5, where n is the position of the term in the sequence.
2. 5, -5/3, 5/9, -5/27 ...
The pattern in this sequence is that each term alternates between multiplying the previous term by -1/3 and dividing the previous term by 3. So, the explicit formula for this sequence is: a(n) = (-1/3)^(n-1) * 5, where n is the position of the term in the sequence.
3. 1.25, 5, 20, 80 ....
The pattern in this sequence is that each term is obtained by multiplying the previous term by 4. So, the explicit formula for this sequence is: a(n) = 1.25 * 4^(n-1), where n is the position of the term in the sequence.
4. -4, -24, -144, -864 ...
The pattern in this sequence is that each term is obtained by multiplying the previous term by -6. So, the explicit formula for this sequence is: a(n) = -4 * (-6)^(n-1), where n is the position of the term in the sequence.
5. 4, 12, 36, 108 ...
The pattern in this sequence is that each term is obtained by multiplying the previous term by 3. So, the explicit formula for this sequence is: a(n) = 4 * 3^(n-1), where n is the position of the term in the sequence.
6. 3, -15, 75, -375 ...
The pattern in this sequence is that each term alternates between multiplying the previous term by -5 and dividing the previous term by -3. So, the explicit formula for this sequence is: a(n) = (-5)^(n-1) * 3, where n is the position of the term in the sequence.
7. 2, 4, 8, 16 ...
The pattern in this sequence is that each term is obtained by multiplying the previous term by 2. So, the explicit formula for this sequence is: a(n) = 2^(n-1), where n is the position of the term in the sequence.
8. 3/2, 3/4, 3/8, 3/16 ...
The pattern in this sequence is that each term is obtained by dividing the previous term by 2. So, the explicit formula for this sequence is: a(n) = (3/2) * (1/2)^(n-1), where n is the position of the term in the sequence.
9. -1, -6, -36, -216 ...
The pattern in this sequence is that each term is obtained by multiplying the previous term by -6. So, the explicit formula for this sequence is: a(n) = -1 * (-6)^(n-1), where n is the position of the term in the sequence.
10. 3, 12, 48, 192 ...
The pattern in this sequence is that each term is obtained by multiplying the previous term by 4. So, the explicit formula for this sequence is: a(n) = 3 * 4^(n-1), where n is the position of the term in the sequence.
Remember, when using these explicit formulas, the term number (position) starts from 1, so be careful with the indexing when plugging in the value of n to determine specific terms in the sequences.