The Geometric Sequence Formula, Explained

A geometric sequence (also called a geometric progression) is a list of numbers where each term is found by multiplying the previous term by the same fixed number. That fixed number is called the common ratio, written r. Once you know the first term and the ratio, a single formula lets you jump straight to any term without writing out the whole list.

The geometric sequence formula

The formula for the nth term is a_n = a_1 times r to the power (n minus 1). Here a_1 is the first term, r is the common ratio, and n is the position of the term you want (1 for the first term, 2 for the second, and so on). The exponent is n minus 1, not n, because the first term has not been multiplied by r at all yet.

• a_n is the term you are solving for (the term in position n).
• a_1 is the first term of the sequence.
• r is the common ratio, equal to any term divided by the one before it.
• n is the position number, always a positive whole number.

To find the common ratio from a known sequence, divide any term by the term immediately before it. For 5, 15, 45, 135, the ratio is 15 divided by 5, which is 3, and 45 divided by 15 is also 3, confirming r equals 3.

A worked example, step by step

Suppose the sequence starts 4, 12, 36, 108, ... and you want the 7th term.

1. Identify the first term: a_1 equals 4.
2. Find the common ratio: 12 divided by 4 equals 3, so r equals 3.
3. Set the position you want: n equals 7.
4. Plug into the formula: a_7 equals 4 times 3 to the power (7 minus 1), which is 4 times 3 to the power 6.
5. Evaluate the power: 3 to the power 6 equals 729.
6. Multiply: 4 times 729 equals 2916. So the 7th term is 2916.

Notice you never had to write out terms 5 and 6. The exponent (n minus 1) does the heavy lifting, which is exactly why the formula is so useful for large positions. This shortcut mirrors the way the compound interest formula reuses a constant multiplier over and over.

How the common ratio shapes the sequence

The value of r decides everything about how the sequence behaves. When r is bigger than 1 the terms grow quickly, a pattern known as exponential growth. When r is between 0 and 1 the terms shrink toward zero. A negative r makes the signs alternate between positive and negative. The reference chart below shows the same starting term of 2 under several ratios.

When r equals 1 every term is identical, so the sequence is technically geometric but completely flat. When r equals 0 the sequence collapses to zero after the first term, which is why most definitions require r to be nonzero.

Recursive formula vs explicit formula

The formula a_n = a_1 times r to the power (n minus 1) is the explicit formula: it jumps straight to any term using only its position. There is a second way to describe the same sequence called the recursive formula, which defines each term in terms of the one before it. The recursive form is a_1 = (first term) and a_n = r times a_(n minus 1) for n of 2 or more.

Both forms describe the identical sequence; they just answer different questions. Use the explicit formula when you want the 50th term directly. Use the recursive formula when you are generating terms one after another, for instance in a spreadsheet or a short program. Many textbooks ask you to translate between the two, so it helps to see them side by side.

Finding a missing term when you know two others

A common exam question gives you two non-adjacent terms and asks for the ones in between. Suppose a_1 equals 3 and a_4 equals 24, and you need the common ratio and the missing middle terms.

1. Write the known term with the formula: a_4 equals a_1 times r to the power 3, so 24 equals 3 times r cubed.
2. Divide both sides by 3: r cubed equals 8.
3. Take the cube root: r equals 2.
4. Fill the gaps by multiplying: a_2 equals 3 times 2 equals 6, and a_3 equals 6 times 2 equals 12.
5. Check the result: the sequence 3, 6, 12, 24 indeed multiplies by 2 each step.

The general trick is the same whenever two terms are known: their ratio equals r raised to the power of the gap between their positions. Solve for r first, then walk forward or backward one step at a time.

Adding up the terms: the geometric series

Often you want the sum of the first n terms rather than a single term. That sum is called a geometric series, and it has its own closed formula: S_n = a_1 times (1 minus r to the power n) divided by (1 minus r), valid whenever r is not equal to 1.

For example, add the first 5 terms of 3, 6, 12, 24, 48. Here a_1 equals 3, r equals 2, and n equals 5. Then S_5 equals 3 times (1 minus 2 to the power 5) divided by (1 minus 2), which is 3 times (1 minus 32) divided by (-1), or 3 times 31, giving 93. A quick check by hand confirms 3 plus 6 plus 12 plus 24 plus 48 equals 93.

If r is between -1 and 1, the terms shrink so fast that even an infinite number of them adds to a finite value, given by a_1 divided by (1 minus r). This idea underpins many results in finance and physics, much like the steady decay you see in a half-life calculation.

Geometric vs arithmetic sequences

It is easy to mix up the two main sequence types. The difference is simple: arithmetic sequences add a constant, geometric sequences multiply by a constant. If you are comparing them side by side, the arithmetic sequence formula guide walks through the matching add-based version.

Where geometric sequences show up in real life

Geometric sequences are not just a textbook exercise. Any quantity that changes by the same percentage each period follows one, because a percentage change is really multiplication by a constant factor. A 4 percent yearly increase means multiplying by 1.04 every year, and a 10 percent yearly decline means multiplying by 0.9. The table below collects the most common real-world cases and the ratio each one uses.

Growth cases (r greater than 1) line up with the exponential growth model, while decay cases (r between 0 and 1) match the half-life pattern. Recognizing the constant multiplier is the single skill that lets you switch between the everyday story and the formula.

Common mistakes to avoid

• Using n instead of n minus 1 in the exponent. The first term is a_1 times r to the power 0, which is just a_1. Forgetting the minus 1 shifts every answer one term too far.
• Adding instead of multiplying. If consecutive terms share a constant difference rather than a constant ratio, you have an arithmetic sequence, not a geometric one.
• Dropping a negative ratio. When r is negative, keep the sign through the power. An even exponent yields a positive term and an odd exponent yields a negative term.
• Applying the sum formula when r equals 1. The division by (1 minus r) is undefined there, so the sum is simply n times a_1 instead.

Good to know

The common ratio can be a fraction, a decimal, or even an irrational number; the formula works the same way. Geometric sequences show up far beyond math class: compounding savings, population models, radioactive decay, and the brightness steps on a screen are all geometric. Once you spot a constant multiplier, you can predict any future value instantly with a_n equals a_1 times r to the power (n minus 1).