Math Buddyv1.0

Recursive equation calculator

Give a starting term and a rule. MathBuddy builds the sequence term by term up to the one you need.

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What this page covers

  • Finding a chosen term from a starting value and a rule that uses the previous term.
  • Rules that use the two previous terms, with two starting values.
  • Turning an arithmetic recursive rule into a formula in nn.
  • Not covered here: finding a formula from a list of terms alone (use the explicit formula calculator), and sequences defined by words or pictures, which you can still type and try.

How to enter the problem

  • Type it or paste it. Write the starting term, then the rule with subscripts, for example a_1 = 2, a_n = 3a_(n-1) - 1, and the term you want.
  • 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

A sequence starts with a1=5\displaystyle a_1 = 5 and follows an=an−1−3\displaystyle a_n = a_{n-1} - 3. Find a6\displaystyle a_6.

Answer

Verified

The answer is

a6=−10a_6 = -10

Explanation

  1. Read the rule
    Each term is the previous term minus 3.
  2. Build the terms in order
    a2=2a_2 = 2, a3=−1a_3 = -1, a4=−4a_4 = -4, a5=−7a_5 = -7.
  3. Take one more step
    a6=a5−3=−7−3=−10a_6 = a_5 - 3 = -7 - 3 = -10

Example 2

Problem

A sequence has a1=1\displaystyle a_1 = 1, a2=3\displaystyle a_2 = 3 and an=an−1+2an−2\displaystyle a_n = a_{n-1} + 2a_{n-2}. Find a6\displaystyle a_6.

Answer

Verified

The answer is

a6=43a_6 = 43

Explanation

  1. Read the rule
    Each new term is the term just before it plus twice the term two places back.
  2. Find the third and fourth terms
    a3=3+2(1)=5a_3 = 3 + 2(1) = 5 and a4=5+2(3)=11a_4 = 5 + 2(3) = 11.
  3. Find the fifth term
    a5=11+2(5)=21a_5 = 11 + 2(5) = 21
  4. Find the sixth term
    a6=21+2(11)=43a_6 = 21 + 2(11) = 43

Example 3

Problem

Write a formula in n\displaystyle n for the sequence a1=7\displaystyle a_1 = 7, an=an−1+5\displaystyle a_n = a_{n-1} + 5.

Answer

Verified

The answer is

an=5n+2a_n = 5n + 2

Explanation

  1. Recognise the pattern
    Adding the same number each time makes an arithmetic sequence with first term 77 and common difference 55.
  2. Count the steps
    To reach ana_n from a1a_1 you add 5 exactly n−1n - 1 times, so an=7+5(n−1)a_n = 7 + 5(n-1).
  3. Simplify
    7+5n−5=5n+27 + 5n - 5 = 5n + 2. Check: n=1n = 1 gives 77.

Common mistakes

  • Taking one step too many. In the example, applying the rule 5 times instead of 4 gives 365365 instead of 122122.
  • Using the term number nn in place of the previous term, as if an=3n−1a_n = 3n - 1.
  • In a two-term rule, mixing up which earlier term gets the coefficient: in an=an−1+2an−2a_n = a_{n-1} + 2a_{n-2} the 2 multiplies the older term.
  • Adding the difference nn times instead of n−1n - 1 times when writing a formula, which gives 5n+75n + 7 instead of 5n+25n + 2.

Checks, assumptions and limits

  • Check by hand: list the terms in a small table, one row per nn, and confirm each row follows from the row above.
  • The rule needs a starting value. Without a1a_1 (and a2a_2 for a two-term rule) the sequence is not fixed.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

It gives each term using the term or terms before it, together with a starting value. To reach a later term you apply the rule step by step.

For rules that add or multiply by a fixed number, yes. Ask for a formula in n and the steps show how it is found.

Yes. The steps list the terms in order, so you can follow each one back to the rule.

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