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
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
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).
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.