nth Partial Sum Calculator Online

The nth Partial Sum Calculator calculates the total of the first ‘n’ terms of a sequence. This function is vital in fields such as engineering, economics, and science, where quick and accurate summation of series is critical.

Formula of nth Partial Sum Calculator

Below are the formulas for calculating the nth partial sum for different sequence types:

Arithmetic Sequence

For an arithmetic sequence, use this formula:

S_n = (n/2) * (2a + (n – 1)d)

Where:

• n is the number of terms
• a is the first term
• d is the common difference

Geometric Sequence

For a geometric sequence, use this formula:

S_n = a * (1 – r^n) / (1 – r) for r not equal to 1

Where:

• n is the number of terms
• a is the first term
• r is the common ratio

Harmonic Sequence

For a harmonic sequence, the nth partial sum is:

S_n = sum of (1/k) for k from 1 to n

Table of Common Terms

Here is a table with common values for partial sums in arithmetic, geometric, and harmonic sequences:

Sequence Type	Term Count (n)	First Term (a)	Common Difference/Ratio (d/r)	nth Partial Sum (S_n)
Arithmetic	5	2	3	35
Geometric	4	1	2	15
Harmonic	3	–	–	1.8333

This table helps users directly apply these values in their calculations without manual computation.

Example of nth Partial Sum Calculator

Consider an arithmetic sequence where the first term ‘a’ is 5, the common difference ‘d’ is 3, and we calculate the sum of the first 4 terms (n=4):

S_4 = (4/2) * (2 * 5 + (4 – 1) * 3) = 2 * (10 + 9) = 38

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