A series is a summation of the terms in a sequence. For the sequence 2, 4, 6, 8, the series can be written as 2+4+6+8. If the terms are given by a rule, it's usually more compact to write it using sigma notation. The sequence above is described by the explicit rule an=2n. The series can be written as n=1∑42n. Here, by substituting n with integers 1 (under Σ) through 4 (above Σ) the individual terms of the series are obtained, and the sigma itself indicates that between each term is a plus sign. This alternative notation is useful if the series contains a large number of terms, and especially useful when the sequence is infinite. Then, the infinity symbol is written above the sigma.
n=1∑∞2nThe summation of the terms in an arithmetic sequence is called an arithmetic series. If the sequence is short enough, such as 1,3,5,7, it's straightforward to calculate the sum: s=1+3+5+7=16.
However, if the sequence is longer, it can be tedious to add the terms by hand. In that case, the formula for an arithmetic sum can be used.For a finite arithmetic sequence given by an=a1+(n−1)d, where a1 is the first term, d is the common difference, and n is the number of terms, the sum of all terms, Sn, can be calculated using the following formula.
Sn=2n(a1+an)
Now, there are 9 fives, and the sum is 9⋅5. Here, 5 is the mean of the first and last term, which means that it can be written as 2a1+an. This is multiplied by 9, which is the number of terms, n, leading to the formula
Sn=n⋅2a1+an=2n(a1+an).Determine the sum of all positive integers between 1 and 1000.