Geometric Sequence Calculator

A geometric sequence multiplies every term by the same fixed number. The nth term is aₙ = a₁ · rⁿ⁻¹ and the sum of the first n terms is Sₙ = a₁(1 − rⁿ) / (1 − r). Example: starting at 3 with a common ratio of 2, the 10th term is 1,536 and the first 10 terms add up to 3,069.

Before you start, confirm that your sequence really is geometric. Divide each term by the one before it. If every ratio is the same number, this page is the right tool. If every difference is the same instead, the sequence is arithmetic and grows in a straight line, so use the Arithmetic Sequence Calculator. If each term is the sum of the two before it, you have the Fibonacci pattern and want the Fibonacci Calculator.

How a geometric sequence works

Every geometric sequence is defined by exactly two numbers: the first term a₁ and the common ratio r. From those two values you can reach any term without listing the ones before it, because each step multiplies by r. Getting to term 2 takes one multiplication, term 3 takes two, and term n takes n − 1. That is why the exponent in the formula is n − 1 rather than n.

The middle identity is the partial sum, valid for every ratio except r = 1. The right-hand identity is the infinite sum, and it is valid only when the absolute value of r is smaller than 1. When r = 1 the sequence is constant and the sum is simply n × a₁, because the partial sum formula would divide by zero.

Worked example: 3, 6, 12, 24, and so on

Take a₁ = 3, r = 2 and n = 10, the values loaded in the calculator above. First the nth term. Ten terms means nine multiplications by 2, so a₁₀ = 3 × 2⁹ = 3 × 512 = 1,536. Then the sum. Substituting into the partial sum formula gives S₁₀ = 3 × (1 − 2¹⁰) / (1 − 2) = 3 × (1 − 1,024) / (−1) = 3 × 1,023 = 3,069. Because |r| = 2 is larger than 1, there is no infinite sum: the terms keep growing and the total runs away.

The full sequence is 3, 6, 12, 24, 48, 96, 192, 384, 768, 1,536. Adding those ten numbers by hand also gives 3,069, which is exactly the point of the closed form: it replaces ten additions with one line of arithmetic. Push n to 20 and the closed form still answers instantly, giving a₂₀ = 1,572,864 and S₂₀ = 3,145,725, numbers that would take a while to reach by hand.

Term-by-term table for a₁ = 3, r = 2

Every value below comes from the two formulas above. Notice that each running sum is exactly the next term minus 3, which is minus a₁, a pattern unique to r = 2.

Second worked example: a shrinking sequence

Now take a₁ = 100 and r = 0.5, so the sequence runs 100, 50, 25, 12.5, 6.25 and keeps halving. The 10th term is 100 × 0.5⁹ = 0.195313, and the sum of the first ten terms is 100 × (1 − 0.5¹⁰) / 0.5 = 199.804688. Because |r| = 0.5 is below 1, this series converges: no matter how many terms you add, the running total never passes S∞ = 100 / (1 − 0.5) = 200. It only gets closer. After 10 terms you are within 0.2 of the limit; after 20 terms you are within 0.0002.

That gap between the partial sum and the limit is exactly a₁rⁿ / (1 − r), which is why a ratio close to 1 converges so slowly. With r = 0.9 the same starting value of 100 has an infinite sum of 1,000, but the first ten terms only reach 651.32156, less than two thirds of the way there.

Third worked example: finding r from two terms

Homework often gives you two scattered terms instead of the first one. Suppose the 3rd term is 20 and the 7th term is 320. Between them lie 7 − 3 = 4 multiplications, so 20 × r⁴ = 320, which means r⁴ = 16 and r = 2. Step back to the start with a₁ = 20 / r² = 20 / 4 = 5. From there the sequence is 5, 10, 20, 40, 80, 160, 320, and asking for the 12th term gives a₁₂ = 5 × 2¹¹ = 10,240 with a partial sum of S₁₂ = 5 × (2¹² − 1) = 20,475. Switch the calculator to the two-term mode and it runs all of these steps for you.

How the ratio changes the outcome

The table below starts every sequence at a₁ = 100 and varies only the common ratio, showing the 10th term and the sum of the first 10 terms. It makes the split between shrinking and growing sequences obvious.

Moving the ratio from 1.1 to 1.5 barely changes the second or third term, yet the 10th term grows sixteen-fold. That is the defining feature of geometric growth: small differences in the ratio compound into enormous differences later.

Infinite sums when |r| is below 1

The infinite geometric series is the one piece of this topic that surprises people: an endless list of positive numbers can still add up to a finite total. The condition is simply that the terms shrink fast enough, which happens exactly when the absolute value of r is under 1. The table starts every series at a₁ = 100 and compares the true limit against the sum of the first ten terms.

The classic textbook use is turning a repeating decimal into a fraction. Write 0.333 repeating as 0.3 + 0.03 + 0.003 and you have a₁ = 0.3 with r = 0.1, so the sum is 0.3 / 0.9 = 1/3. The same argument on 0.9 + 0.09 + 0.009 gives 0.9 / 0.9 = 1, which is the honest proof that 0.999 repeating equals exactly one.

How to use this geometric sequence calculator

1. Pick your mode. Use "I know a₁ and r" when the problem hands you the first term and the ratio. Use "I know two terms" when it only gives you two scattered terms.
2. Enter the first term a₁. It can be negative or a decimal, but never zero.
3. Enter the common ratio r. Divide any term by the one before it to find it. Negative ratios and fractions are fine.
4. Enter n, the position of the term you want. Whole numbers from 1 to 10,000.
5. Read the result. The headline shows aₙ. The tiles below give the partial sum, the infinite sum (or "diverges"), the first term and the percentage change per step.
6. Scroll for the working. The rule card shows the explicit and recursive formulas, the table lists the first 25 terms with a running sum, and the step-by-step panel walks through the algebra.

Who this calculator is for

• Algebra 2 and precalculus students checking nth term and series homework before it is due.
• SAT, ACT and AP Calculus test takers practicing convergence questions where the |r| < 1 condition is the whole point.
• Teachers and tutors generating clean example sequences with known sums.
• Finance and science users modeling anything that changes by a fixed percentage per period, from compounding balances to radioactive decay.
• Programmers sizing exponential backoff delays, buffer-doubling strategies or binary tree node counts.

Key terms explained

• First term (a₁): the value the sequence starts at. It scales everything, since every later term is a multiple of it.
• Common ratio (r): the fixed multiplier between consecutive terms, found by dividing any term by the one before it.
• Explicit formula: aₙ = a₁rⁿ⁻¹, which jumps straight to any term.
• Recursive formula: aₙ = aₙ₋₁ × r, which describes each step but forces you to walk the list.
• Partial sum (Sₙ): the total of the first n terms, always counted from term 1.
• Convergence: the property that the partial sums approach a fixed limit, which for a geometric series happens exactly when |r| < 1.
• Geometric mean: the middle term of any three consecutive terms equals the square root of the product of its neighboring terms. For 3, 6, 12 that is the square root of 36, which is 6.

Geometric versus arithmetic growth side by side

Both sequences below start at 100 and run for 10 terms. The arithmetic one adds 10 per step; the geometric one multiplies by 1.10 per step. They look almost identical at first and separate sharply later.

Both start at the same place and even share the same second term, yet the geometric total is 143.74 higher after only ten steps. Stretch the same comparison to 30 steps and the geometric version is many times larger. Compounding is the reason a fixed percentage return outruns a fixed dollar return over a long horizon, and the Compound Interest Calculator applies exactly this arithmetic to a balance.

What changes the result the most

• The common ratio, by a wide margin. It sits in an exponent, so changing it from 1.05 to 1.10 multiplies the 20th term by about 2.42 rather than adding 5 percent.
• The term position n. Each extra term multiplies the answer by r again, which is why long sequences explode or vanish.
• The first term, which only scales the result. Doubling a₁ doubles every term and doubles both sums, never changing the shape.
• Whether |r| crosses 1. This single threshold decides between a finite infinite sum and no sum at all, and it does so abruptly: r = 0.999 converges, r = 1.001 does not.
• The sign of r. A negative ratio makes terms alternate, so the partial sums oscillate above and below the limit instead of rising to it.

Practical tips

• To sum terms 11 through 20, compute S₂₀ and subtract S₁₀. For 3, 6, 12 that is 3,145,725 − 3,069 = 3,142,656.
• To find which position holds a given value, use n = log(aₙ / a₁) / log(r) + 1. For a₁ = 3, r = 2 and aₙ = 3,072 that gives exactly 11.
• Check any middle term against its neighboring terms: it should equal the square root of their product.
• A ratio written as a percentage change converts easily. Growth of 5 percent per period means r = 1.05; a 15 percent yearly loss means r = 0.85.
• If your ratio is negative, expect the running sum to bounce. With a₁ = 8 and r = −0.5 the sixth partial sum is 5.25 while the limit is 5.333333.

Real-world geometric sequences

Percentage change per period is the signature of a geometric sequence, and it appears everywhere. A vehicle that loses 15 percent of its value each year has r = 0.85, so a $30,000 car is worth $25,500 after one year, $21,675 after two, $18,423.75 after three and $13,311.16 after five. A drug with a four-hour half-life leaves 200 mg, 100 mg, 50 mg, 25 mg and 12.5 mg in the bloodstream at four-hour marks, a sequence with r = 0.5 that the Half-Life Calculator models directly. A ball dropped from 10 feet that rebounds to 60 percent of its previous height travels 10 + 2 × 10 × 0.6 / (1 − 0.6) = 40 feet in total before it stops, which is the infinite sum formula answering a physical question. And the famous grain-on-a-chessboard puzzle is nothing but a₁ = 1 with r = 2 and n = 64, whose sum is 2⁶⁴ − 1, or 18,446,744,073,709,551,615.

Limitations and assumptions

• Results are shown as decimals rounded to 6 places, not exact fractions, so a ratio such as 1/3 displays as 0.333333.
• Very large or very small values switch to scientific notation and carry ordinary floating-point rounding beyond about 15 significant digits.
• The ratio must be a real number. Complex ratios and sequences whose ratio changes from step to step are outside the scope.
• The two-term mode returns the principal real root. When the position gap is even and both values are positive, a negative ratio of the same magnitude also fits, so check the sign of your other terms.
• The tool assumes your list really is geometric; it cannot tell you that your data merely looks exponential.

Which calculator should you use?

This page answers "what is the nth term and the sum of a sequence that multiplies?" A sister tool fits better for neighboring questions. Use the Arithmetic Sequence Calculator when the gap between terms is constant rather than the ratio. Use the Compound Interest Calculator when the sequence is a money balance and you want dollars, deposits and years instead of a₁, r and n. Use the Exponent Calculator when you only need to evaluate a single power such as 2¹⁰. Use the Half-Life Calculator for decay stated in half-lives rather than a ratio. Use the Fibonacci Calculator when each term is the sum of the two before it, which is neither arithmetic nor geometric.