What is the valuve of the sum of
1+3+5+7+9+11 ...+ 1997 +1999 ?
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
The sum of the members of a finite arithmetic progression is called an arithmetic series.
The sum S of the first n values of a finite sequence is given by the formula:
Sn=(n/2)*[2*a1+(n-1)*d]
In this case:
n=1000
a1=1
d=2
Sn=(1000/2)*[2*1+(1000-1)*2]
Sn=500*[2+999*2]
Sn=500*[2+1998]
Sn=500*2000
Sn=1,000,000
In this case:
1 is first term
3 is second term
5 is third term
7 is fourth term
etc
1999 is thousandth term
because that in your progression
n=1000
To find the value of the sum of the series 1+3+5+7+9+11...+1997+1999, we can use the formula for the sum of an arithmetic series.
An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, the difference between each pair of consecutive terms is 2.
The formula to find the sum of an arithmetic series is:
Sn = (n/2) * (a + L)
where Sn is the sum of the series, n is the number of terms, a is the first term, and L is the last term.
First, we need to find the number of terms in the series. The series starts with 1, increases by 2 each time, and ends at 1999. We can find the number of terms by subtracting the first term from the last term, dividing by the common difference (2), and adding 1:
n = (L - a)/d + 1
= (1999 - 1)/2 + 1
= 1998/2 + 1
= 999 + 1
= 1000
Next, we can substitute the values into the formula:
Sn = (n/2) * (a + L)
= (1000/2) * (1 + 1999)
= 500 * 2000
= 1,000,000
Therefore, the sum of the series 1+3+5+7+9+11...+1997+1999 is 1,000,000.