AP & GP Calculator

Find the nth term and finite sum of an arithmetic or geometric progression, with a term preview and worked steps.

Enter your values

Change any field. The result is recalculated on this device.

Formula, example and assumptions
Term 10 (aₙ)21

Define the progression

Progression type

This is term 1, not term 0. Positive, negative and zero values are allowed.

Add this amount to get the next term. Use a negative difference for a decreasing AP.

Choose the term and preview

A whole number from 1 to 1,000,000. Calculates both aₙ and the sum from term 1 through term n.

Show up to 50 terms, never more than n. The preview limit does not change the sum.

Your result

Local calculation
Term 10 (aₙ)21

The nth term is one position in the progression; the finite sum adds every term from 1 through n. Changing the preview length does not change either answer.

Progression type
AP — add a difference
Common difference (d)
2
Term number / number of terms (n)
10
See how this is calculated

Result breakdown

Sum of the first 10 terms (Sₙ)
120
Number of terms in the finite sum
10
View worked ap calculation and first 10 terms

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Worked AP calculation and first 10 terms
StepWorking

The finite sum includes all 10 terms; the preview shows only the first 10. Terms are indexed from 1. Negative differences, ratios and terms are allowed. A zero GP ratio means the first term followed by zeroes. Working uses 80 significant digits; results show up to 12. Inputs retain approximately 15–17 significant digits. Results outside the finite numeric range produce a range error, not infinity or a false zero. This is not an infinite-series sum.

Check the inputs and method

These are your current inputs, not the worked example further down the page.

Progression type
AP — add a difference
First term (a₁)
3
Common difference (d)
2
Term number / number of terms (n)
10
First terms to show
10

Method

AP: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2. GP: aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1, or n × a₁ when r = 1. With r = 0, term 1 is a₁, subsequent terms are zero and every finite sum is a₁.

Read the worked example and assumptions

How this calculation works

A sequence lists terms; a series adds them. Choose arithmetic progression for a constant step and geometric progression for a constant multiplier. The result separates the requested nth term from the sum of all n terms, then shows the formula substitution and a bounded preview so you can check how the sequence develops.

AP: aₙ = a₁ + (n − 1)d; Sₙ = n(a₁ + aₙ)/2. GP: aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1, or n × a₁ when r = 1. With r = 0, term 1 is a₁, subsequent terms are zero and every finite sum is a₁.

Worked example

This example uses illustrative inputs. Your current result is shown above.

AP with a₁ = 3, d = 2 and n = 10: the terms begin 3, 5, 7, 9; a₁₀ = 3 + 9×2 = 21, and S₁₀ = 10×(3 + 21)/2 = 120. Switching to GP with a₁ = 3 and r = 2 gives a₁₀ = 3×2⁹ = 1536 and S₁₀ = 3×(1 − 2¹⁰)/(1 − 2) = 3069.

Assumptions and limits

  • The first term has index 1. The same n specifies the requested term and the number of terms included in the finite sum.
  • An AP adds a fixed difference; a GP repeatedly multiplies by a fixed ratio. Zero first terms and zero GP ratios use this recurrence convention even where a ratio of successive zero terms would be undefined.
  • First term, difference and ratio are finite numbers between −10¹² and 10¹². n is a whole number from 1 to 1,000,000; the preview is limited to 50 terms.
  • Arithmetic uses 80 significant digits and results show up to 12. Numeric inputs preserve approximately 15–17 significant digits. A result beyond the finite numeric range, or a nonzero result too small to represent, produces a range error.
  • This calculator finds finite sums only. It does not assume that an infinite geometric series converges.

Questions people ask

What is the difference between an AP and a GP?

An AP adds the same difference each time, such as 3, 5, 7 with d = 2. A GP multiplies by the same ratio, such as 3, 6, 12 with r = 2. A percentage change must first be converted to a multiplier: 10% growth uses r = 1.1.

Can the terms or ratio be negative?

Yes. Negative differences can make an AP decrease. A negative GP ratio alternates the sign of nonzero terms. With r = −1, the terms alternate a₁ and −a₁; an even number of terms sums to zero and an odd number sums to a₁.

What happens with r = 0 or r = 1?

With r = 0, the first term stays a₁ and every later term is zero. With r = 1, all terms equal a₁ and the finite sum is n×a₁. These cases are evaluated separately to avoid an undefined division or an ambiguous zero-power expression.

Does showing only ten terms mean only ten terms are added?

No. The preview limit affects only the displayed term list. The finite sum always includes term 1 through term n. A range error means the requested result is outside the supported numeric range; it does not mean the sequence itself is mathematically undefined.

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