How to Find Sums of a Series
When a pattern of numbers must be totaled, you can use an arithmetic or geometric series to find the sum. An arithmetic series has an equal difference between each number in the sequence, while a geometric series has a difference that decreases or increases with an equal proportion over the entire sequence. You can sum the series by hand -- that is, writing out all the terms and adding them up with a calculator -- or you can use a special formula to save yourself time.
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Instructions
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Determine if the series is arithmetic or geometric. An arithmetic series has a constant difference between each term, such as 1, 4, 7, 10 and13, while a geometric series has equal ratios between terms. That is, the ratio of the first term to the second term is the same as the ratio of the second term to the third, and the same as the ratio of the third term to the fourth.
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Determine if the series is finite or infinite. If the last term is given, then it is finite; otherwise, it is infinite.
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Find the ratio between terms if it is a geometric sequence. That is, divide a term by the previous term, and call this ratio "r." If the value of r is outside the range of -1 and 1, the sum cannot be calculated.
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Subtract the ratio "r" from 1, divide the first term in the sequence by this value to find the sum of an infinite geometric sequence. In the series 32, 16, 8, 4 ..., the sum would be (32)/(1 - (1/2)) = 64.
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Multiply the first term by (1 - r^n), then divide by (1 - r), where n is the number of terms in the geometric series. This is the sum of a finite geometric series. In the series 1, 1/2, 1/4, 1/8, the sum would be (1)(1 - (1/2)^4)/(1- 1/2) = 1.875.
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Multiply n by 1/2, then multiply by the sum of the first and last terms to get the arithmetic sum of a finite series. For example, summing the even numbers between 2 and 100 would be (1/2)(50)(2+100) = 2,550.
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References
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