HexMathv1.0.3

Sequence calculator: finding the pattern

Enter the first few terms. HexMath finds the pattern and writes the rule for the nth term.

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What this page covers

  • Arithmetic patterns, where the same number is added each time.
  • Geometric patterns, where each term is multiplied by the same number.
  • Quadratic patterns, where the differences themselves go up by a fixed amount.
  • Not covered here: sums of many terms (use the partial sum calculator), and lists with no clear rule, where several patterns fit the same terms.

How to enter the problem

  • Type it or paste it. List at least 4 terms separated by commas and say what you want, the nth term or a term number.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Find the 15th term of 20,17,14,11,…\displaystyle 20, 17, 14, 11, \dots

Answer

Verified

The answer is

a15=−22a_{15} = -22

Explanation

  1. Look at the differences
    Each term is 3 less than the one before, so the common difference is −3-3.
  2. Write the rule
    an=20+(n−1)(−3)a_n = 20 + (n-1)(-3)
  3. Put in n=15n = 15
    a15=20+14(−3)=20−42=−22a_{15} = 20 + 14(-3) = 20 - 42 = -22

Example 2

Problem

Find the rule for the nth term of 2,6,18,54,…\displaystyle 2, 6, 18, 54, \dots

Answer

Verified

The answer is

an=2⋅3n−1a_n = 2 \cdot 3^{n-1}

Explanation

  1. Look at the ratios
    6÷2=36 \div 2 = 3, 18÷6=318 \div 6 = 3, 54÷18=354 \div 18 = 3. Each term is 3 times the one before.
  2. Write the geometric rule
    First term 22, ratio 33, so an=2⋅3n−1a_n = 2 \cdot 3^{n-1}.
  3. Test it
    n=4n = 4 gives 2⋅27=542 \cdot 27 = 54, which matches.

Example 3

Problem

Find the rule for the nth term of 3,6,11,18,27,…\displaystyle 3, 6, 11, 18, 27, \dots

Answer

Verified

The answer is

an=n2+2a_n = n^2 + 2

Explanation

  1. First differences
    3,5,7,93, 5, 7, 9. These are not constant.
  2. Second differences
    2,2,22, 2, 2. A constant second difference means the rule is quadratic, with n2n^2 coefficient 2÷2=12 \div 2 = 1.
  3. Remove the n2n^2 part
    Subtract n2n^2 from each term: 3−1,6−4,11−9,…3 - 1, 6 - 4, 11 - 9, \dots gives 2,2,2,…2, 2, 2, \dots
  4. Put the parts together
    What remains is the constant 22, so an=n2+2a_n = n^2 + 2.

Common mistakes

  • Using nn in place of n−1n - 1 in the arithmetic rule, which gives 4n+74n + 7 instead of 4n+34n + 3 for the example.
  • Calling a pattern arithmetic when the differences change. Check two or three differences before writing a rule.
  • Writing a geometric rule with exponent nn instead of n−1n - 1, which makes every term 3 times too big in 2,6,18,542, 6, 18, 54.
  • Stopping at the first differences for a quadratic pattern. The second differences show the n2n^2 part.

Checks, assumptions and limits

  • Check by hand: put n=1,2,3n = 1, 2, 3 into the rule and make sure it gives back the terms you started with.
  • A few terms never prove a pattern. The rule found is the simplest one that fits the terms given.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

4 or more is safest. Two or three terms fit many different rules, so more terms make the pattern clear.

Yes. Ask for the next term, or for a term number such as the 20th, and the steps show the rule first.

The steps look at differences and ratios. If the second differences are constant, the rule is quadratic and is found the same way.

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Last updated: · HexMath