Geometric series sum:
formula and worked example

What a geometric series total describes

A geometric series adds up numbers that each equal the previous one multiplied by a fixed ratio. A ratio above one means growth, such as a savings amount that doubles each period or a viral message that is forwarded to twice as many people each hour. A ratio between zero and one means decay, such as a bouncing ball that rises to a fraction of its previous height.

The total matters when you want the cumulative effect: how much is saved in all after several periods, how many people have been reached altogether, or how much was repaid in total. The individual terms alone tell you only the last step, not the running sum.

Adding six or seven terms by hand is doable, but twenty is not, and the closed form handles any number with the same effort.

The formula and what to feed it

The total equals the first term times the quantity ratio-to-the-power-of-terms minus one, divided by ratio minus one.

• First: the starting term, 3 in the example.
• Ratio: the constant multiplier between neighbouring terms, 2 in the example.
• Terms: how many terms are added, counting the first, 6 in the example.

The ratio is found by dividing any term by the one before it. If the ratio is less than one, the formula still works and gives a total that levels off as the number of terms grows.

Worked example: 3 doubled across six terms

Start at 3, double each time and add the first six terms.

1. 1Raise the ratio to the number of terms: 2^6 = 64.
2. 2Subtract one: 64 − 1 = 63.
3. 3Divide by ratio minus one: 63 ÷ (2 − 1) = 63.
4. 4Multiply by the first term: 3 × 63 = 189.

Listing the terms confirms it: 3 + 6 + 12 + 24 + 48 + 96 = 189. Note that the last term, 96, is more than half of the total, which is typical when the ratio is 2.

How to read a geometric total

With a ratio of 2, the sum always equals the next term in the pattern minus the first term: the term after 96 would be 192, and 192 − 3 = 189. This gives a rapid mental check.

For ratios above one, the final few terms dominate the sum. Cutting the number of terms by one from six to five does not take away a sixth of the total: it removes 96, about half. This is why exponential growth schedules behave so differently from linear ones.

For a ratio below one, the total is bounded. An endless series with a ratio of one half and first term 3 never exceeds 6, which equals first divided by (1 minus ratio). That limit applies only when the ratio is between -1 and 1.

Sensitivity to the first term

The first term multiplies everything else, so the sum is directly proportional to it. A 10% rise in the first term gives a 10% rise in the total, regardless of the ratio or term count.

The ratio is far more sensitive. Moving it from 2 to 2.2 with the same first term and count increases the total much more than 10%, because the ratio is compounded across all six terms.

Traps in geometric series sums

• Using a ratio of exactly 1. The denominator becomes zero, so the formula fails; the correct sum is first multiplied by the number of terms.
• Mixing up the number of terms with the exponent. For six terms the power is 6, not 5, in the form used here, which already accounts for the first term.
• Treating percentage growth as a ratio. A 5% increase per period means a ratio of 1.05, not 0.05.
• Applying it to a sequence where the difference, not the ratio, is constant, which is arithmetic and needs a different rule.

A savings example

Someone saves 500 in the first month and increases the amount by 10% each month for 12 months. Will this method give the total saved?

1. 1Identify the ratio: a 10% rise each month gives 1.1.
2. 2Raise it to the power of 12 and subtract one.
3. 3Divide by 0.1 and multiply by 500.

Because every month is 10% above the last, the series is geometric. Doing the same work with a flat 500 each month would instead be 500 × 12.