Arithmetic Sequence Calculator

Arithmetic Sequence is evaluated from First Term, Common Difference and Number of Terms. The calculation reports nth Term, Sum of n Terms and Last Term.

Results

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About the Arithmetic Sequence Calculator

Arithmetic Sequence is treated here as a quantitative relation between First Term, Common Difference and Number of Terms and nth Term, Sum of n Terms, Last Term and Average Term.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
aₙ = a₁ + (n - 1)d
Sₙ = n/2 x (2a₁ + (n - 1)d) = n/2 x (a₁ + aₙ)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

aₙ = a₁ + (n - 1)d
Sₙ = n/2 x (2a₁ + (n - 1)d) = n/2 x (a₁ + aₙ)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Odd Numbers Sum (1 + 3 + 5 + ... + 99)

Inputs

first_term: 1 common_d: 2 n_terms: 50
nth Term: 99. Sum of n Terms: 2,500. Last Term: 99. Average Term: 50

With First Term = 1, Common Difference = 2 and Number of Terms = 50 as the stated inputs, the result is nth Term = 99, Sum of n Terms = 2,500 and Last Term = 99. Each value corresponds to the declared output fields.

Example 2: Stacking Cans — Triangular Arrangement

Inputs

first_term: 1 common_d: 1 n_terms: 15
nth Term: 15. Sum of n Terms: 120. Last Term: 15. Average Term: 8

With First Term = 1, Common Difference = 1 and Number of Terms = 15 as the stated inputs, the result is nth Term = 15, Sum of n Terms = 120 and Last Term = 15. Each value corresponds to the declared output fields.

Example 3: Depreciation — Straight-Line Method

Inputs

first_term: 50000 common_d: -8000 n_terms: 6
nth Term: 10,000. Sum of n Terms: 180,000. Last Term: 10,000. Average Term: 30,000

With First Term = 50,000, Common Difference = -8,000 and Number of Terms = 6 as the stated inputs, the result is nth Term = 10,000, Sum of n Terms = 180,000 and Last Term = 10,000. Each value corresponds to the declared output fields.

Example 4: Seating in an Auditorium

Inputs

first_term: 20 common_d: 3 n_terms: 25
nth Term: 92. Sum of n Terms: 1,400. Last Term: 92. Average Term: 56

With First Term = 20, Common Difference = 3 and Number of Terms = 25 as the stated inputs, the result is nth Term = 92, Sum of n Terms = 1,400 and Last Term = 92. Each value corresponds to the declared output fields.

Common Use Cases

  • Find the nth term of a sequence like 3, 7, 11, 15...
  • Calculate sum of first n terms of an AP
  • Find how many terms to reach a target value
  • Solve annuity-like linear payment schedules