How do I use the sigma notation to express the series : 1+5+25+125+625+3125?
First try to make a relation/rule between the different terms:
1=50
5=51
25=52
125=53
625=54
3125=55
So you are really summing
50+51+52+53+54+55
So you would end up with:
i=5
Σ 5i
i=0
To use sigma notation to express the series 1+5+25+125+625+3125, follow these steps:
Step 1: Identify the pattern.
In this series, each term is obtained by multiplying the previous term by 5. The first term is 1, and each subsequent term is 5 times the previous term.
Step 2: Write the general term.
The general term can be represented as 5^(n-1), where n is the position or index of each term in the series.
Step 3: Determine the range.
For this series, the range is from n = 1 to n = 6, as there are 6 terms in the series.
Step 4: Use sigma notation.
The sigma notation is used to represent a sum of terms. To express this series using sigma notation, write:
∑(from n = 1 to n = 6) of 5^(n-1).
Step 5: Simplify the expression.
To simplify the expression further, you can expand the sigma notation:
∑(from n = 1 to n = 6) of 5^(n-1)
= 5^(1-1) + 5^(2-1) + 5^(3-1) + 5^(4-1) + 5^(5-1) + 5^(6-1)
= 5^0 + 5^1 + 5^2 + 5^3 + 5^4 + 5^5
= 1 + 5 + 25 + 125 + 625 + 3125
Therefore, the series 1+5+25+125+625+3125 can be expressed in sigma notation as ∑(from n = 1 to n = 6) of 5^(n-1).
To express the series 1+5+25+125+625+3125 using sigma notation, you can analyze the pattern in the terms.
The terms of this series are increasing based on powers of 5. The first term, 1, can be written as 5^0. The second term, 5, can be written as 5^1. The third term, 25, can be written as 5^2. The fourth term, 125, can be written as 5^3. The fifth term, 625, can be written as 5^4. And the sixth term, 3125, can be written as 5^5.
Now we can express the terms using an index variable, let's call it "n." The value of n will range from 0 to 5, as there are 6 terms in the series. For each value of n, the corresponding term can be written as 5^n.
To write the series in sigma notation, we can use the sigma symbol (∑) and specify the starting and ending values of n, and the expression for each term:
The sigma notation for the series 1+5+25+125+625+3125 is:
∑(from n=0 to 5) 5^n
This notation indicates that we sum up the terms of 5^n for n starting from 0 and going up to 5.
Evaluating this summation will give you the value of the series.