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
Most Common FAQs
A1: Yes, the calculator is equipped to handle sequences of virtually any length with defined input parameters.
A2: For r=1, the formula changes to simply multiplying the first term ‘a’ by the number of terms ‘n’.
A3: The calculator is highly accurate, utilizing precise formulas to ensure each calculation adheres to true mathematical principles.