Pick the kind of sequence, type the starting number and the step, and say which term you want. The calculator gives that term, the total of all terms up to it, and writes out the beginning of the sequence so you can see the pattern.
Choose arithmetic, geometric or Fibonacci.
Type the first number and the difference or ratio.
Type which term you want.
Read the term, the sum and the list of terms.
Arithmetic: nth term: a + (n − 1) × d
Arithmetic: sum of n terms: n × (first + last) ÷ 2
Geometric: nth term: a × r^(n − 1)
Geometric: sum of n terms: a × (1 − r^n) ÷ (1 − r)
Fibonacci: F(n) = F(n − 1) + F(n − 2), with F(0) = 0 and F(1) = 1
A number sequence calculator works with lists of numbers that follow a fixed rule. You tell it the rule and it tells you any term you ask for, without writing out all the terms before it. It also adds up the terms for you. Three kinds of sequence are covered here: arithmetic, geometric and Fibonacci.
In school these come up as AP and GP in Class 10 and Class 11. Outside the classroom they appear more often than people expect. A salary that rises by a fixed amount every year is an arithmetic sequence. Money growing at a fixed percentage is a geometric sequence. The Fibonacci pattern shows up in puzzles, coding interviews and even in the way petals and seeds are arranged in flowers.
1. Decide the type: is the same number being added each time, or multiplied each time?
2. Note the first number of the sequence.
3. Find the common difference (subtract two neighbours) or the common ratio (divide two neighbours).
4. Decide which term you need, for example the 20th.
5. Use the nth-term formula for that type.
6. If the total is needed, use the sum formula.
When the gap between neighbours never changes, the sequence is arithmetic. To reach the nth term you start at the first term and take n − 1 steps, which gives a + (n − 1) × d. The sum has a neat shortcut: pair the first term with the last, the second with the second-last, and so on. Every pair adds to the same total, so the sum is the number of terms times the average of the first and last terms.
When each term is the previous one times a fixed number, the sequence is geometric. The nth term is a × r^(n − 1). Because of the repeated multiplication the numbers climb very quickly when r is more than 1, which is the idea behind compound interest. When r lies between −1 and 1 the terms shrink towards zero, and even an endless list of them adds up to a fixed number, a ÷ (1 − r).
The Fibonacci sequence does not use a fixed step or ratio. Every term is made by adding the two terms just before it, starting from 0 and 1. The numbers grow slowly at first and then very fast: the 50th is already more than 1,200 crore. The ratio of one term to the previous one moves closer and closer to about 1.618, a value known as the golden ratio.
Ramesh joins a firm in Pune at a salary of 30,000 rupees a month with a fixed raise of 2,500 rupees every year. He wants to know his monthly salary in the 10th year.
Formula for the nth term: a(n) = a + (n − 1) × d
Put in the numbers: a(10) = 30000 + (10 − 1) × 2500 = 52500
Sum of the first 10 terms: S = n × (first + last) ÷ 2 = 10 × (30000 + 52500) ÷ 2 = 412500
Answer: Term number 10 52500; Sum of the first 10 terms 412500; The sequence begins 30000, 32500, 35000, 37500, 40000, 42500, 45000, 47500, 50000, 52500
Using n instead of n − 1 in the nth-term formula, which gives the next term by mistake.
Mixing up the common difference and the common ratio.
Forgetting that a negative difference makes the sequence fall.
Using the geometric sum formula when the ratio is 1, which leads to division by zero.
Counting the Fibonacci terms from 1 when the question starts them from 0.
AP and GP problems in Class 10 and 11 maths.
Working out a salary with fixed yearly increments.
Seeing how savings grow at a fixed yearly percentage.
Coding practice questions on Fibonacci numbers.
Bank and aptitude exam questions on number series.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence grows by adding a fixed number. A geometric sequence grows by multiplying by a fixed number, so it rises (or shrinks) much faster.
How do I find the common difference?
Subtract any term from the one after it. In 4, 9, 14, 19 the common difference is 9 − 4 = 5.
How do I find the common ratio?
Divide any term by the one before it. In 3, 6, 12, 24 the common ratio is 6 ÷ 3 = 2.
Does the Fibonacci sequence start at 0 or 1?
Both are used. This calculator starts with F0 = 0 and F1 = 1, so F10 = 55.
Can a geometric sequence have a sum that never ends?
Yes, when the ratio is between −1 and 1. The terms keep getting smaller and the total settles at a ÷ (1 − r).