Writing the rule in summation notation
-2+4-8+16-32
The rule for the given pattern can be written in summation notation as:
∑((-1)^(n+1) * 2^n), where n starts from 1 and goes to infinity.
To write the given series -2+4-8+16-32 in summation notation, we need to determine the pattern of the terms.
Looking at the series, we can observe that each term is obtained by multiplying the previous term by -2. Therefore, the expression can be written as:
∑((-2)^n) * (-2)^n, where n represents the position of the term in the series.
In mathematical notation, the summation notation for the given series is:
∑((-2)^n) * (-2)^n, where n ranges from 0 to infinity.
To write the given rule in summation notation, we need to express it as a sequence with a common pattern. Looking at the given sequence -2, 4, -8, 16, -32, we can observe that each term alternates between being positive and negative, and the absolute value of each term doubles from the previous one.
We can start by expressing the sequence as a function of `n`, where `n` represents the position of each term in the sequence. The pattern can be described as follows:
- For even values of `n`, the term is positive and given by 2^n.
- For odd values of `n`, the term is negative and given by -2^n.
Next, we can define the summation notation using the pattern we identified. The summation notation for the given sequence can be written as:
∑ (-1)^(n+1) * 2^n, where `n` ranges from 1 to infinity.
In this notation, the term (-1)^(n+1) ensures that the signs alternate between positive and negative, and 2^n represents the doubling pattern observed. Note that the summation starts at `n = 1` to match the pattern in the given sequence, and it is open-ended as it continues to infinity.