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Methods for Calculating Partial Sums

Published in Partial Sums 2 mins read

How to Find a Partial Sum

A partial sum is the sum of a portion of a sequence. To find a partial sum, you need to know the sequence and how many terms to include in the sum.

The method for finding a partial sum depends on the type of sequence.

1. Arithmetic Series

For an arithmetic series (where the difference between consecutive terms is constant), a simple formula exists:

  • Formula: S<sub>n</sub> = ∑<sub>i=1</sub><sup>n</sup> a<sub>i</sub> = n(a<sub>1</sub> + a<sub>n</sub>) / 2

    • Where:
      • S<sub>n</sub> is the partial sum of the first n terms.
      • a<sub>1</sub> is the first term.
      • a<sub>n</sub> is the nth (last) term.
      • n is the number of terms.

  • Example: Find the partial sum of the first 10 terms of the arithmetic sequence 2, 5, 8, 11...

    • a<sub>1</sub> = 2
    • a<sub>10</sub> = 2 + (10 - 1) * 3 = 29 (common difference is 3)
    • S<sub>10</sub> = 10(2 + 29) / 2 = 155

2. Other Sequences

For sequences that are not arithmetic, there's no single universal formula. The approach depends on the pattern of the sequence. Methods might include:

  • Direct Summation: Add the terms directly. This is feasible for short sequences.
  • Formula Derivation: If the sequence has a clear pattern (e.g., a geometric sequence, a series defined by a function), derive a formula for the partial sum.
  • Computational Tools: For complex sequences, use software like Python or mathematical software packages (Mathematica, MATLAB) to calculate the sum. Python's sum() function is particularly useful.

3. Infinite Series

For infinite series, you typically find the limit of the sequence of partial sums, which is the series' sum, if the limit converges to a finite value. Determining whether an infinite series converges is a separate but related mathematical topic.

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