SeqLab

Geometric Sequence Calculator

aₙ = a₁ · rⁿ⁻¹.

First term a₁

Common ratio r

Term index n

aₙ (nth term)

486

aₙ = a₁ · rⁿ⁻¹

Sₙ (sum of n terms): 728

What you entered

First term a₁: 2Common ratio r: 3Term index n: 6

1
nth term: a₁ · rⁿ⁻¹
2 × 3^5= 486
2
Sum of first n terms
2 × (1 − 3^6) ÷ (1 − 3)= 728

Result

aₙ (nth term): 486

Starting at 2 with common ratio 3, term 6 is 486, and the first 6 terms sum to 728.

Geometric sequences: nth term and sum

Each term multiplies the previous one by a constant ratio. Ratios below 1 shrink the sequence and its sum converges.

First term a1	Ratio r	Terms n	nth term	Sum of n terms
2	3	6	486.0000	728.0000
1	2	10	512.0000	1023.0000
3	2	12	6144.0000	12285.0000
5	1.5	7	56.9531	160.8594
100	0.5	8	0.7813	199.2188

Row two is the doubling sequence, and its sum of 1,023 is always one less than the next term - which is why a byte holding 8 doubling bits maxes out at 255 rather than 256. The last row halves each time, and its sum is closing on 200: a geometric series with a ratio below 1 converges to a finite total no matter how many terms you add, which is the resolution of Zeno's paradox. Above a ratio of 1 the growth is exponential, which is why compound interest, population growth and viral spread are all geometric.

A constant multiplier, not a constant step

Where arithmetic sequences add the same amount each time, geometric sequences multiply by the same ratio r — 2, 6, 18, 54… multiplies by 3 every step, which is why growth accelerates so much faster.

Where it shows up

Compound interest, population growth, and viral spread are all geometric processes — this is the same mathematical structure behind 'doubling every period' scenarios.

Frequently asked questions

A bacteria colony doubles every hour starting with 500. How many after 10 hours?

Geometric sequence with a₁ = 500 and r = 2. After 10 hours: a₁₁ = 500 × 2¹⁰ = 500 × 1,024 = 512,000 bacteria. (Note: hour 0 is term 1, so hour 10 is term 11.)

How is a geometric sequence related to compound interest?

Compound interest is a geometric sequence where the common ratio r = (1 + rate). A $1,000 investment at 5% annual interest gives the sequence 1000, 1050, 1102.50, 1157.63… — each term is the previous multiplied by 1.05. The compound interest calculator handles this directly.

What happens when the common ratio is between 0 and 1?

The sequence shrinks toward zero. A bouncing ball that reaches 80% of its previous height follows r = 0.8: heights of 10, 8, 6.4, 5.12 meters. The infinite sum converges to a₁/(1 − r) = 10/0.2 = 50 meters total distance.