Utilnaro

Math formulas

Sequence formulas

Calculate terms and sums of arithmetic and geometric sequences, including convergent infinite series.

Formulas and calculators

Arithmetic sequence

Calculate the nth term or the sum of the first n terms of an arithmetic sequence, or work backwards to the first term, the common difference, or the index.

aₙ = a₁ + (n - 1) × r; Sₙ = n × (a₁ + aₙ) / 2

Calculated quantities

a₁ - first term · r - common difference · n - term index · aₙ - nth term · Sₙ - sum of the first n terms

Open formula

Geometric sequence

Calculate the nth term, the sum of the first n terms, or the sum of a convergent infinite geometric series, and work backwards to a₁, q, or n.

aₙ = a₁ × qⁿ⁻¹; Sₙ = a₁ × (1 - qⁿ) / (1 - q)

Calculated quantities

a₁ - first term · q - common ratio · n - term index · aₙ - nth term · Sₙ - sum of the first n terms · S - sum of the infinite series

Open formula

How to choose the right formula

Arithmetic sequence

aₙ = a₁ + (n - 1) × r; Sₙ = n × (a₁ + aₙ) / 2
When to use it
In an arithmetic sequence every term differs from the previous one by the same value r, so the nth term grows linearly and the partial sum is the number of terms times the average of the first and the last term.
Calculated quantities
a₁ - first term (1) · r - common difference (1) · n - term index (1) · aₙ - nth term (1) · Sₙ - sum of the first n terms (1)
Worked example: Twelfth term of a sequence
An arithmetic sequence starts with a₁ = 5 and has a common difference r = 3. Calculate a₁₂.
Open formula

Geometric sequence

aₙ = a₁ × qⁿ⁻¹; Sₙ = a₁ × (1 - qⁿ) / (1 - q)
When to use it
In a geometric sequence every term is the previous one multiplied by the same ratio q, so the nth term grows exponentially; the partial sum needs a separate formula for q = 1, and the whole infinite series has a finite sum only for |q| < 1.
Calculated quantities
a₁ - first term (1) · q - common ratio (1) · n - term index (1) · aₙ - nth term (1) · Sₙ - sum of the first n terms (1) · S - sum of the infinite series (1)
Worked example: Eighth term of a sequence
A geometric sequence starts with a₁ = 3 and has a common ratio q = 2. Calculate a₈.
Open formula

What to check before calculating

Assumptions and conditions

  • The difference r between consecutive terms is constant for the whole sequence.
  • The index n is a natural number greater than or equal to 1; there is no term number 0 and no term number 2.5.
  • The common difference can only be derived from a₁ and aₙ when n is at least 2, because a₁ alone carries no information about r.
  • The index can only be derived when r is different from zero; for r = 0 every term equals a₁.
  • The sum formula covers the first n terms in order, starting from a₁.
  • The ratio q between consecutive terms is constant, different from zero, and the first term a₁ is also different from zero.
  • The index n is a natural number greater than or equal to 1.
  • The partial sum formula with (1 - q) in the denominator is valid only for q ≠ 1; for q = 1 the sum equals n × a₁.

Common mistakes

  • Using n instead of n - 1 in the explicit formula and shifting the whole sequence by one term.
  • Entering a non-integer or non-positive index; the sequence has no term number 0 and no term number 3.5.
  • Using the sum formula with the first term instead of the last term, that is Sₙ = n × 2a₁ / 2.
  • Trying to derive the common difference from a single term, or the index when r = 0.
  • Using the exponent n instead of n - 1 and returning the next term of the sequence.
  • Applying Sₙ = a₁ × (1 - qⁿ) / (1 - q) for q = 1, which divides by zero instead of giving n × a₁.
  • Summing an infinite series with |q| ≥ 1, where the partial sums grow without a limit.
  • Entering a₁ = 0 or q = 0, which does not define a geometric sequence.