Geometric sequence nth term:
a × r^(n − 1) and what it models

Multiplying instead of adding

Here every term is the previous one multiplied by the same number, the common ratio r. Starting at 3 with r = 2 gives 3, 6, 12, 24, 48. With r = 0.5 starting at 100 you get 100, 50, 25, 12.5, a halving sequence.

To find r, divide any term by the one before it. The ratio can be a fraction, negative or greater than 1. A negative ratio makes the terms alternate in sign: 7, −14, 28, −56.

The contrast with the additive kind is pace. Add 3 each time and 20 steps gets you to 62. Multiply by 2 each time and 20 steps gets you past a million. This is why geometric patterns describe doubling, compound interest, depreciation and radioactive decay.

The nth term formula

The exponent is n − 1 because the first term has not been multiplied at all, the second once, and so on. Write 3 × 2¹⁰ = 3,072 and you have found the 11th term.

As with other sequences, you may be given two terms that are not neighbours. If the 2nd term is 12 and the 5th is 96, three steps separate them, so r³ = 96 ÷ 12 = 8 and r = 2. The first term is then 12 ÷ 2 = 6. Taking a cube root here is the geometric counterpart of dividing by the number of steps in the additive case.

Finding the ratio and the position

Given two terms, find the ratio by dividing. For the sequence 6, 18, 54, the ratio is 18 ÷ 6 = 3, and the 10th term is 6 × 3⁹ = 6 × 19,683 = 118,098.

To find which term a value is, solve a × r^(n − 1) = value. For 3, 6, 12, … and the value 768, divide by 3 to get 256, which is 2⁸. So n − 1 = 8 and n = 9.

When the target is not a neat power, you need logarithms: n = log(target ÷ a) ÷ log(r) + 1. A calculator handles that in one step.

Link to compound growth

A deposit that earns a fixed percentage each period follows a geometric sequence with r = 1 + rate. Money of ₹10,000 at 8% a year becomes 10,000 × 1.08 after one year, 10,000 × 1.08² after two and so on.

The position shift is worth noticing. The amount after n years is term n + 1, because the opening amount is the first term. This is why growth examples often seem to be off by one when read against the formula.

Sum of the terms

For 3, 6, 12, … the first 10 terms add to 3 × (1,024 − 1) ÷ 1 = 3,069. When |r| is below 1 the terms shrink toward zero, and the infinite sum is a ÷ (1 − r). For 4, 4/3, 4/9, … with r = 1/3, the total is 4 ÷ (2/3) = 6.

Shrinking by a fixed percentage

Falling values follow the same pattern with r below 1. A machine bought for ₹50,000 that loses 15% of its value each year has r = 0.85. After 3 years it is worth 50,000 × 0.85³ = ₹30,706.25. After 4 years it is worth ₹26,100.31, still above zero, because a percentage loss never reaches nothing.

Halving works the same way. A substance that halves in a fixed period has r = 0.5. After 6 periods, 64 units become 64 × 0.5⁶ = 1 unit. The amounts shrink quickly at first and slowly afterwards, which is the shape of any exponential decay.

What the graph looks like

Plot a geometric sequence and the points do not line up. For r > 1 they curve upward, gently at first and then steeply. For r between 0 and 1 they fall away and flatten near zero. Plot the logarithm of the terms instead and the points become a straight line, which is the standard way to test whether data is growing exponentially.

The doubling case is a good feel for the speed. Start with 1 and double ten times: you get 1,024. After 20 doublings it is over a million, and after 30 it is over a billion. This is why a modest-looking growth rate, compounded over many periods, surprises people.

Common mistakes

• Using n instead of n − 1 as the exponent.
• Confusing r = 1.08 with r = 0.08 in a growth problem.
• Treating a pattern like 1, 4, 9, 16 as a multiplying one; its ratios are not equal.
• Rounding the ratio early, which makes a high-n term wander widely from the true value.
• Applying the sum formula with r = 1, which divides by zero.

A sequence calculator lets you test a ratio quickly and watch how fast the terms move, especially for large n where hand calculation is awkward.