Math Buddyv1.0

Explicit formula calculator

Enter a sequence or two known terms. MathBuddy writes the formula for the nth term, step by step.

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What this page covers

  • Explicit formulas for arithmetic sequences, from a list of terms or from two known terms.
  • Explicit formulas for geometric sequences, including negative ratios.
  • Using an explicit formula to find any term directly.
  • Not covered here: building terms one at a time from a rule (use the recursive formula calculator), and sequences with no fixed difference or ratio.

How to enter the problem

  • Type it or paste it. List the terms separated by commas, or give two terms such as a_3 = 11 and a_8 = 31.
  • 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 explicit formula for an arithmetic sequence with a3=11\displaystyle a_3 = 11 and a8=31\displaystyle a_8 = 31.

Answer

Verified

The answer is

an=4n−1a_n = 4n - 1

Explanation

  1. Find the common difference
    From a3a_3 to a8a_8 is 5 steps, and the value rises by 31−11=2031 - 11 = 20, so d=20÷5=4d = 20 \div 5 = 4.
  2. Find the first term
    a1=a3−2d=11−8=3a_1 = a_3 - 2d = 11 - 8 = 3
  3. Write the formula
    an=3+4(n−1)=4n−1a_n = 3 + 4(n-1) = 4n - 1
  4. Test it
    n=8n = 8 gives 32−1=3132 - 1 = 31, which matches.

Example 2

Problem

Find the explicit formula for 5,−10,20,−40,…\displaystyle 5, -10, 20, -40, \dots

Answer

Verified

The answer is

an=5(−2)n−1a_n = 5(-2)^{n-1}

Explanation

  1. Find the ratio
    −10÷5=−2-10 \div 5 = -2 and 20÷(−10)=−220 \div (-10) = -2, so the ratio is r=−2r = -2.
  2. Write the geometric formula
    an=a1rn−1=5(−2)n−1a_n = a_1 r^{n-1} = 5(-2)^{n-1}
  3. Test it
    n=4n = 4 gives 5(−8)=−405(-8) = -40, which matches. The sign flips each term because the ratio is negative.

Example 3

Problem

A sequence has explicit formula an=3n2−n\displaystyle a_n = 3n^2 - n. Find a7\displaystyle a_7.

Answer

Verified

The answer is

a7=140a_7 = 140

Explanation

  1. Substitute the term number
    Put n=7n = 7
    a7=3(7)2−7a_7 = 3(7)^2 - 7
  2. Work out the square first
    3⋅49−7=147−7=1403 \cdot 49 - 7 = 147 - 7 = 140

Common mistakes

  • Writing an=a1+dna_n = a_1 + dn instead of a1+d(n−1)a_1 + d(n-1). For the example this gives 6−4n6 - 4n, which is wrong at n=1n = 1.
  • Dividing the change by the wrong number of steps. From a3a_3 to a8a_8 there are 5 steps, not 8.
  • Dropping the brackets around a negative ratio: 5⋅−2n−15 \cdot -2^{n-1} is not the same as 5(−2)n−15(-2)^{n-1}.

Checks, assumptions and limits

  • Check by hand: put n=1n = 1 and n=2n = 2 into the formula and confirm you get the first two terms.
  • The term number nn is a whole number starting at 1. Some books start at n=0n = 0, which shifts the formula.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

An explicit formula gives any term straight from its position n. A recursive formula needs the previous term first.

Yes. For an arithmetic sequence two terms are enough. Type them with their positions, for example a_3 = 11 and a_8 = 31.

Yes. The steps find the common ratio and write the formula as the first term times the ratio to the power n minus 1.

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