How to Find Finite Series

Finding the sum of a finite series depends on the type of series. The most common types are arithmetic and geometric series. Let's explore how to find their sums.

Arithmetic Series

An arithmetic series is a sequence where the difference between consecutive terms is constant (this difference is called the common difference). To find the sum of a finite arithmetic series, use the following formula:

S_n = n(a₁ + a_n)/2

Where:

• S_n is the sum of the first n terms.
• n is the number of terms.
• a₁ is the first term.
• a_n is the last (nth) term.

Study.com provides a clear explanation of this formula.

Example: Find the sum of the arithmetic series 2, 5, 8, 11, 14.

Here, n = 5, a₁ = 2, and a₅ = 14. Plugging these values into the formula:

S₅ = 5(2 + 14)/2 = 40

Therefore, the sum of the series is 40.

Geometric Series

A geometric series is a sequence where each term is found by multiplying the previous term by a constant value (called the common ratio). The sum of a finite geometric series is calculated using this formula:

S_n = a₁(1 - rⁿ) / (1 - r)

Where:

• S_n is the sum of the first n terms.
• a₁ is the first term.
• r is the common ratio.
• n is the number of terms. The importance of correctly identifying n is emphasized in Sal's demonstration of the formula derivation.

This formula is detailed on websites like Varsity Tutors and CK-12 Foundation. Note that this formula is valid only when r ≠ 1.

Example: Find the sum of the geometric series 1, 3, 9, 27, 81.

Here, a₁ = 1, r = 3, and n = 5. Substituting into the formula:

S₅ = 1(1 - 3⁵) / (1 - 3) = 121

The sum of the series is 121.

Other Finite Series

For other types of finite series, there might not be a single, readily available formula. However, a formula can always be found to fit any finite sequence of numbers, as explained on Mathematics Stack Exchange. This often involves higher-level mathematical techniques like polynomial interpolation. For practical purposes, calculators or software such as the Omnicalculator sum of series calculator can be used to evaluate many finite series.