Math & Statistics

Geometric Sequence Calculator

Calculate the nth term, a selected indexed term, finite partial sum, growth factor, total change, comparison-sum gap, and selected-term gap for a geometric sequence with a non-unit common ratio.

First termCommon ratioNumber of termsSelected term indexEntered comparison sumEntered term comparison

Term at entered count-

Term at selected index-

Sum through entered term count-

Nth term divided by first-

Nth term minus first term-

Partial sum minus comparison-

Selected term minus entered reference-

Decision view

Geometric term curve and cumulative series area

Geometric term curve and cumulative series areaIndexed term growth and the running finite sum share one timeline, with selected and final terms marked separately.

Exact scenario comparisonCommon ratio changes while all other entered assumptions remain constant.

Common ratio	Term at entered count	Term at selected index	Sum through entered term count	Nth term divided by first	Nth term minus first term	Partial sum minus comparison	Selected term minus entered reference

Period-by-period detail

Complete geometric-sequence table

Every indexed term and cumulative total updates with first term, ratio, and term count.

How to use Geometric Sequence Calculator

Enter the first term and a common ratio other than one.
Enter the number of terms and a selected index.
Read individual-term growth separately from the cumulative sum.
Use the table and curves to inspect every indexed value.

Calculator guide

Understanding Geometric Sequence Calculator

A geometric sequence changes by multiplication rather than by a fixed additive step. This calculator separates an individual term from the cumulative finite sum and shows how the common ratio compounds across indices.

Multiplicative change Each step multiplies by r.

Index starts at one The exponent is n - 1.

Term and sum differ One value is not the cumulative series.

Curves update together Term and cumulative paths share the same inputs.

Calculation method

How the calculation works

Raise the entered common ratio to indexed powers for terms and apply the finite geometric-series formula for the cumulative sum. In the Geometric Sequence Calculator, the live scenario varies common ratio and tracks term at entered count while the remaining results preserve the reconciliation path. Raise the common ratio to index minus one for each term. Sum the first n terms with the finite geometric-series formula.

Detailed calculation process

Compound the ratio by index and reconcile the cumulative sum

The defaults start at 100, multiply each term by 1.08, select term 8, and extend the sequence through term 20.

General formula: a_n = a_1 r^(n-1); S_n = a_1(1-r^n)/(1-r), r != 1 The exponent counts how many multiplicative steps separate a term from the first. The sum formula combines all indexed terms without adding them one by one.

What each symbol means

a_1 First term in the sequence's value unit.

r Common multiplicative ratio, unitless.

n Number or index of terms, measured as terms.

a_n Value of the nth term in the original value unit.

S_n Cumulative sum through term n in the value unit.

j Entered selected term index.

Worked substitution with the default inputs

1. Count multiplicative steps: n - 1 = 20 - 1 = 19 The first term has exponent zero, so term 20 applies the ratio 19 times.

2. Calculate the twentieth term: a_20 = 100 x 1.08^19 = 431.570106 The twentieth term is about 4.3157 times the first.

3. Calculate the selected term: a_8 = 100 x 1.08^7 = 171.382427 Seven compounding steps separate term 8 from term 1.

4. Calculate the finite sum: S_20 = 100(1 - 1.08^20)/(1 - 1.08) = 4,576.196430 The closed form reconciles the cumulative total of all twenty terms.

5. Compare totals and references: 4,576.196430 - 5,000 = -423.803570; 171.382427 - 300 = -128.617573 The final term increased by 331.570106 from the first while both entered comparisons remain higher.

The defaults produce term 20 = 431.570, term 8 = 171.382, and a 20-term sum of 4,576.196, with a total growth factor of 4.3157.

Growth anatomy

Compare the term path with the accumulated path

The cumulative curve rises by exactly the current term at every index.

Term curve Shows the value added at each index.

Cumulative curve Shows all contributions through each index.

Selected marker Locates the entered j.

Final reconciliation Matches a_n and S_n to result cards.

Worked situations

Practical examples

A ratio of 1.08 means each term is 8% larger than the preceding term.
Term 20 is about 4.316 times term 1.
The cumulative sum 4,576.196 is much larger than the final term alone.

Better inputs

Useful tips

Confirm whether your index starts at one.
Distinguish a sequence term from a series sum.
Use logarithmic or normalized views when growth spans many orders of magnitude.

Before relying on the result

Limitations and common mistakes

The closed form shown excludes r = 1, which would produce a constant sequence.
Large term counts or ratios can overflow numeric limits.
Real systems may not sustain one constant ratio indefinitely.

Reference

Key terms

Common ratio: Multiplier connecting each term to the previous term.
Nth term: Value at a specified sequence index.
Partial sum: Total of all terms from 1 through n.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why is the exponent n minus one?

Term 1 uses no multiplication, term 2 uses one, and term n uses n - 1.

What if the ratio is below one?

Positive ratios below one produce a decaying sequence and a slower-growing partial sum.

What if the ratio is negative?

Terms alternate signs; interpretation and chart scaling require additional care.

Why is r = 1 excluded?

The displayed sum formula divides by 1 - r; a constant sequence instead has S_n = n a_1.