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nth Partial Sum Calculator Online

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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)

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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 TypeTerm Count (n)First Term (a)Common Difference/Ratio (d/r)nth Partial Sum (S_n)
Arithmetic52335
Geometric41215
Harmonic31.8333

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

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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

Q1: Can the nth Partial Sum Calculator handle sequences with any number of terms?

A1: Yes, the calculator is equipped to handle sequences of virtually any length with defined input parameters.

Q3: How accurate are the calculations from this calculator?

A3: The calculator is highly accurate, utilizing precise formulas to ensure each calculation adheres to true mathematical principles.

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