Geometric Sequence Calculator

Enter three numbers and the calculator does the rest. The first term sets where the sequence starts, the common ratio sets how fast it grows or shrinks, and the number of terms sets how far it goes.

1. Enter the first term. It can be positive, negative or a decimal.
2. Enter the common ratio. Use a number above 1 to grow, a number between 0 and 1 to shrink, or a negative number for terms that alternate in sign.
3. Enter how many terms you want. The nth term is the last one in the list.
4. Leave the term list box ticked to see each term from the first to the nth.
5. Press Calculate, or press Enter. The nth term appears first and large, with the sum, the infinite sum and the ratio below.

A geometric sequence multiplies by the same number at every step. That number is the common ratio, written r. The first term is a1, and the term at position n is a_n = a1 x r^(n-1). The exponent is n - 1 because the first term needs zero multiplications to reach itself.

The sum of the first n terms is S_n = a1 x (1 - r^n) / (1 - r). When r is 1, every term equals a1, so the sum is n x a1. When the absolute value of r is below 1, the terms shrink toward zero and the infinite sum is S = a1 / (1 - r). Our arithmetic sequence calculator covers sequences that grow by adding instead of multiplying.

Take a sequence that starts at 2 with a common ratio of 3, and find the 5th term and the sum of the first five terms.

1. List the terms. Multiply by 3 at each step. The terms are 2, 6, 18, 54, 162.
2. Find the 5th term. a5 = 2 x 3^(5-1) = 2 x 81 = 162.
3. Add the first five terms. S5 = 2 x (1 - 3^5) / (1 - 3) = 2 x (1 - 243) / (-2) = 2 x 121 = 242.
4. Check the infinite sum. The absolute value of 3 is above 1, so the series diverges and has no finite infinite sum.

As a second check, try a shrinking sequence. With a first term of 100 and a ratio of 0.5, the 4th term is 12.5, the first four terms add to 187.5, and the infinite sum is 100 / (1 - 0.5) = 200. The same steps work for any first term and ratio, so try your own numbers in the tool above.

A geometric sequence converges only when the absolute value of the common ratio is below 1. Then each term is smaller than the one before it, the terms close in on zero, and the total of all the terms settles on a finite number. The formula is S = a1 / (1 - r).

Take 100, 50, 25, 12.5 and so on. The ratio is 0.5, so the infinite sum is 100 / (1 - 0.5) = 200. Adding more terms gets closer to 200 but never passes it. The first ten terms already add to about 199.8, which shows how fast a converging series settles down.

When the absolute value of r is 1 or more, the series diverges. A ratio of 1 gives a flat sequence that grows without bound as you add terms. A ratio above 1 grows faster at every step, and a ratio below -1 swings wider in both directions. In each case there is no finite infinite sum, and the calculator says so instead of giving a misleading number.

The most common mistake is using r^n instead of r^(n-1) for the nth term. The first term is a1, not a1 x r, so the exponent is always one less than the position. For the 5th term of a sequence with a ratio of 3, the exponent is 4, not 5.

A second mistake is forgetting the r = 1 case in the sum formula. Plugging r = 1 into S_n = a1 x (1 - r^n) / (1 - r) divides by zero. When the ratio is 1, every term equals the first term, so the sum is simply n x a1. The calculator handles this case for you.

A third mistake is reading the sign of a negative ratio wrong. With a ratio of -2 and a first term of 3, the terms are 3, -6, 12, -24, so the signs alternate and the 4th term is -24, not 24. Check the sign of each term in the term list before you trust the answer.

Compound interest is the most familiar geometric sequence. Money that grows by 5 percent a year multiplies by 1.05 at each step, so a balance of $1,000 becomes $1,050, then $1,102.50, then $1,157.63. The ratio is 1.05 and the nth term formula gives the balance after any number of years. Our compound interest calculator runs the same math with deposits and monthly compounding.

Populations, bacteria and viral spread follow the same pattern while growth stays steady. A quantity that doubles every hour has a ratio of 2, so it grows from 1 to 2 to 4 to 8 and on. Decay works the other way. A medicine that halves in the body every 6 hours has a ratio of 0.5, and the infinite sum formula tells you the total exposure over time. Our exponent calculator helps with the powers that show up in these problems.