Sequence Rule Finder

Intro: Enter a sequence of terms and Mathix will check whether the pattern is arithmetic or geometric, then explain the rule and predict the next term(s). If the list does not fit these simple patterns, the calculator will say so clearly.

Worked example

• The sequence is 3, 7, 11, 15, ... Determine if it is arithmetic or geometric and find aₙ.



---



• We list the given terms: $a_1=3$, $a_2=7$, $a_3=11$, $a_4=15$.



• Check for arithmetic pattern by computing differences: $a_2-a_1 = 7-3=4$, $a_3-a_2=11-7=4$, $a_4-a_3=15-11=4$.



• Since the differences are constant (4), the sequence is arithmetic with common difference $d=4$.



• The general form of an arithmetic sequence is $a_n = a_1 + (n-1)d$.



• Here $a_1=3$ and $d=4$, so $a_n = 3 + (n-1)\cdot4$.



• Simplify: $a_n = 3 + 4n - 4 = 4n - 1$.



• We can check with $n=4$: $a_4 = 4\cdot4-1=15$, which matches the given term.



• Answer: $\boxed{\text{Arithmetic with } d=4,\; a_n=4n-1.}$





• Arithmetic example 3, 7, 11, 15



---



• Next term 19



• Rule: $a_n = 4n - 1$



• Find the differences: $7 - 3 = 4, 11 - 7 = 4, 15 - 11 = 4$.



• The differences are constant, so the sequence is arithmetic with $d = 4$.



• Use $a_n = a_1 + (n - 1)d$ to get $a_n = 3 + (n - 1)4 = 4n - 1$.





• Geometric example 2, 6, 18, 54



---



• Next term 162



• Rule: $a_n = 2 \cdot 3^{n-1}$



• Find the ratios: $\frac{6}{2} = 3,\; \frac{18}{6} = 3,\; \frac{54}{18} = 3$.



• The ratios are constant, so the sequence is geometric with r = 3.



• Use $a_n = a_1 \cdot r^{n-1}$ to get $a_n = 2 \cdot 3^{n-1}$.

How the sequence recognizer works

• The calculator compares consecutive terms to test for a constant difference or a constant ratio. If the differences are constant, it identifies an arithmetic sequence. If the ratios are constant, it identifies a geometric sequence. It then returns the common difference or ratio, the nth-term formula, and the requested next term(s).


• This tool is designed for clear arithmetic and geometric sequences. It can handle positive numbers, negatives, decimals, and fractions when the pattern is exact. If the sequence does not match either pattern, the calculator explains that no simple rule was detected instead of guessing.


• The result shows the detected sequence type, the common difference or ratio, the explicit formula, and the next term(s). When step-by-step mode is enabled, Mathix also shows how the rule was inferred from the input terms.

How to find arithmetic or geometric sequence formulas?

• Enter the known terms of the sequence, separated by commas.


• Choose how many next terms you want predicted.


• Run the calculator to check differences and ratios, then read the identified rule and explanation.

Related

• Sequence Sum Calculator