Number Sequence Calculator

Three patterns, three questions

A number sequence is a list of numbers that follows a rule. Three rules cover most of what you will meet in school and in everyday planning. In an arithmetic sequence you add the same amount every time (2, 7, 12, 17…). In a geometric sequence you multiply by the same amount every time (2, 10, 50, 250…). In the Fibonacci sequence each number is the sum of the previous two (1, 1, 2, 3, 5, 8…). For each one the calculator gives you the term you ask for, the total of all terms up to that point, and the first few terms so you can check you have the right pattern.

Arithmetic: a worked example

Start at 2 and add 5 each time. The 20th term is 2 + (20 − 1) × 5 = 97. The total of the first 20 terms is 20 × (2 + 97) ÷ 2 = 990, which is the number of terms times the average of the first and last. This is the same trick a young Gauss is said to have used to add 1 to 100 in seconds. Real-life examples include saving $5 more each week than the week before, the number of seats in each row of a theatre that grows by a fixed amount, and simple interest, which adds the same dollar amount every period. For regular savings with interest on top, use the Savings Calculator.

Geometric: a worked example

Start at 2 and multiply by 5 each time. The 12th term is 2 × 5¹¹ = 97,656,250, and the sum of the first 12 terms is 122,070,312. Geometric sequences describe anything that grows or shrinks by a percentage: a ratio of 1.05 is 5% growth per step, and a ratio of 0.5 means the quantity halves each time. That makes them the maths behind compound interest, population growth and radioactive decay (see the Half-Life Calculator). Terms get very big very quickly, so if you see an answer in E-notation you can paste it into the Scientific Notation Calculator, and the Big Number Calculator will keep every digit.

When a geometric series never stops growing, and when it settles

If the ratio is between −1 and 1, the terms shrink towards zero and the total of all the infinitely many terms settles on a fixed value. For a first term of 2 and a ratio of 0.5 the sequence runs 2, 1, 0.5, 0.25… and the total approaches 4, which is a ÷ (1 − r). The calculator shows this infinite sum whenever it applies. If the ratio is 1 or more, or −1 or less, the total just keeps growing, so there is no infinite sum.

Fibonacci in practice

The Fibonacci numbers start 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The 10th is 55 and the first ten add up to 143, which is always the number two places further on (F₁₂ = 144) minus one. The ratio of one number to the one before it gets closer and closer to 1.618, the golden ratio, which is why the sequence shows up in spirals of sunflower seeds and pine cones. The calculator keeps every digit, so you can ask for the 100th number (354,224,848,179,261,915,075) or the 1,000th, which has 209 digits.

Finding the rule from a list

If you have a list and are not sure which pattern it follows, subtract each term from the next. If the gaps are always equal, it is arithmetic and that gap is d. If instead dividing each term by the one before always gives the same answer, it is geometric and that answer is r. If neither works, try adding the two terms before each one. For comparing growth, the Percentage Calculator converts a ratio such as 1.08 into a percentage change, and the Statistics Calculator summarises any list of numbers once you have it.

Common questions