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Arithmetic Series

Arithmetic Series

A series such as 3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n. The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.

 

Formula: Formula: Sum = n times the quantity (a₁ + aₙ) divided by 2   or    Formula: (n/2)[2a₁ + (n-1)d]
 

 
Example:

3 + 7 + 11 + 15 + ··· + 99 has a1 = 3 and d = 4. To find n, use the explicit formula for an arithmetic sequence.

We solve 3 + (n – 1)·4 = 99 to get n = 25.

Formula: Sum = 25 times (3 + 99) divided by 2, equals 1275

or  Formula: Sum = (25/2)[2·3 + (25−1)·4] = 1275, showing arithmetic series sum calculation with n=25, a1=3, d=4.

 

See also

Geometric series