rewrite the series in sigma notation 4+16+64+...+256+1024
First, recognize that 4 + 16 + 64 + 256 + 1024 can be rewritten as:
4^1 + 4^2 + 4^3 + 4^4 + 4^5
Now, realize that this is just a sum of 4^n, where the lower limit of n is 1 and the upper limit is 5, as shown above.
Putting this into standard sigma notation yields:
5
Σ 4^n
n=1
To rewrite the series 4 + 16 + 64 + ... + 256 + 1024 in sigma notation, we need to find a general expression for the terms in the series and determine the limits of the summation.
First, let's observe the pattern in the series. Each term is obtained by multiplying the previous term by 4. Starting from the first term, we have:
Term 1 = 4
Term 2 = 4 * 4 = 16
Term 3 = 16 * 4 = 64
Term 4 = 64 * 4 = 256
Term 5 = 256 * 4 = 1024
We can see that each term can be written as 4 raised to the power of (n-1), where n represents the position of the term in the series.
Therefore, the general expression for the terms in the series is given by 4^(n-1).
Next, we need to determine the limits of the summation. The series starts at the first term, which corresponds to n = 1, and ends at the last term, which corresponds to n = 5.
Now, we can rewrite the series in sigma notation:
∑(from n = 1 to 5) 4^(n-1)
Thus, the series 4 + 16 + 64 + ... + 256 + 1024 can be rewritten in sigma notation as ∑(from n = 1 to 5) 4^(n-1).
To rewrite the given series in sigma notation, we can observe that each term is obtained by multiplying the previous term by 4.
We can start by writing the first few terms of the series:
4 + 16 + 64 + ...
To find the general term, we can express each term as a power of 4:
4^1 + 4^2 + 4^3 + ...
Now, we are ready to write the series in sigma notation. Sigma notation uses the Greek letter sigma (∑) to represent the sum of a series. We'll denote the index variable as "n" since it typically represents the position of each term in the series.
The general term of the series is 4^n. To express the sum using sigma notation, we need to specify the range of values for "n".
We start with the first term, n = 1, and the last term is the term before 1024, which is 256. So the range of values for "n" is from 1 to 4.
Finally, we can write the series in sigma notation as:
∑(from n = 1 to 4) 4^n
This represents the sum of the terms where "n" starts from 1 and goes up to 4, and each term is given by 4^n.
We note that the series is a geometric series with a common ratio of r=16/4=4
The first term, i=1, is given by
4=1*4^1=4^i
The second term, i=2, is given by
16=4^2=4^i
...
and the last term, i=5, is given by
1024=4^5=4^i
Therefore the summation is for
4^i for i=1 to i=5.