What is the 20th term of the sequence that begins -2, 4, -8, 16...?
The sequence is defined by $a_n=(-2)^n$, so the 20th term is $a_{20}=(-2)^{20}=\boxed{1048576}$.
To find the 20th term of the sequence, we need to determine the pattern first.
Looking at the given sequence, we can observe that each term is obtained by multiplying the previous term by -2.
So, the pattern can be described as follows:
-2, 4, -8, 16, ...
To find the 20th term, we can use the formula for the n-th term of a geometric sequence:
an = a1 * r^(n-1)
where:
an is the n-th term,
a1 is the first term of the sequence, and
r is the common ratio.
In this case, a1 is -2 (the first term) and r is -2 (the common ratio).
Let's substitute these values into the formula and calculate the 20th term:
a20 = -2 * (-2)^(20-1)
a20 = -2 * (-2)^19
a20 = -2 * 524288
a20 = -1048576
Therefore, the 20th term of the sequence is -1,048,576.
To find the 20th term of the given sequence, we first need to determine the pattern or rule that generates the sequence. Looking at the given sequence, we can notice that each term is obtained by multiplying the previous term by -2. This means the pattern for generating the sequence is multiplying each term by -2.
To calculate the 20th term using this pattern, we can start with the first term (-2) and apply the pattern 19 times since we want the 20th term.
Here are the steps to calculate the 20th term:
1. Start with the first term: -2
2. Apply the pattern by multiplying the previous term by -2 each time:
1st term: -2
2nd term: -2 * -2 = 4
3rd term: 4 * -2 = -8
4th term: -8 * -2 = 16
...
19th term: -2 * -2 * -2 * ... * -2 (19 times)
3. Calculate the 20th term by applying the pattern one more time:
20th term: Previous term * -2
Now, let's calculate the 20th term step by step:
To simplify the calculation, we can observe that multiplying an even number of negative signs results in a positive number. Since we multiply -2 for the pattern 19 times, it means we will end up with a positive 2 in the calculation.
1. Start with the first term: -2
2. Apply the pattern 19 times by multiplying by -2:
20th term = -2 * (-2 * -2 * -2 * ... * -2) (19 times)
20th term = -2 * (-2^19)
Since we know that (-2)^19 = -524,288:
20th term = -2 * (-524,288)
20th term = 1,048,576
Therefore, the 20th term of the given sequence is 1,048,576.