Math Buddyv1.0

Common ratio calculator with steps

Type or scan a sequence. MathBuddy finds the common ratio and shows that every pair of terms agrees.

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What this page covers

  • The common ratio from consecutive terms, including fractional and negative ratios.
  • The ratio from two terms that are not next to each other, such as a1a_1 and a4a_4.
  • A test that the sequence is geometric at all: every ratio of neighbours must be the same.
  • Not covered here: the sum of the terms (use Geometric series calculator) and a formula for the nnth term (use Explicit formula calculator).

How to enter the problem

  • Type the terms separated by commas, or give two terms with their positions, such as a1 = 4 and a4 = 108.
  • 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 common ratio of 5,−15,45,−135\displaystyle 5, -15, 45, -135.

Answer

Verified

The answer is

r=−3r = -3

Explanation

  1. Divide a term by the one before it
    −155=−3\frac{-15}{5} = -3
  2. Repeat for the other pairs
    45−15=−3\frac{45}{-15} = -3 and −13545=−3\frac{-135}{45} = -3.
  3. Confirm
    All three ratios agree, so the sequence is geometric. The signs alternate because the ratio is negative.

Example 2

Problem

A geometric sequence has a1=4\displaystyle a_1 = 4 and a4=108\displaystyle a_4 = 108. Find the common ratio.

Answer

Verified

The answer is

r=3r = 3

Explanation

  1. Write the general term
    an=a1rn−1a_n = a_1 r^{n-1}, so a4=4r3a_4 = 4r^3.
  2. Set up the equation
    4r3=1084r^3 = 108
  3. Solve
    r3=27r^3 = 27, so r=3r = 3. A real cube root has only one value.

Example 3

Problem

A geometric sequence has a2=2\displaystyle a_2 = 2 and a4=18\displaystyle a_4 = 18. Find every possible common ratio.

Answer

Verified

The answer is

r=3,r=−3r = 3, r = -3

Explanation

  1. Relate the two terms
    Two steps separate a2a_2 and a4a_4, so a4=a2r2a_4 = a_2 r^2.
  2. Set up the equation
    2r2=182r^2 = 18, so r2=9r^2 = 9.
  3. Take both square roots
    r=3r = 3 or r=−3r = -3. Both fit: 2,6,182, 6, 18 and 2,−6,182, -6, 18.

Common mistakes

  • Subtracting instead of dividing. 3−2=13 - 2 = 1 is a common difference; the common ratio of 2,3,922, 3, \frac{9}{2} is 32\frac{3}{2}.
  • Dividing in the wrong order. The ratio is a term divided by the term before it, so 23\frac{2}{3} is wrong here.
  • Keeping only the positive root. r2=9r^2 = 9 gives r=3r = 3 or r=−3r = -3.
  • Using the wrong power between distant terms. From a1a_1 to a4a_4 the power is 4−1=34 - 1 = 3, not 4.

Checks, assumptions and limits

  • Multiply back from the first term: 2⋅32=32 \cdot \frac{3}{2} = 3, 3⋅32=923 \cdot \frac{3}{2} = \frac{9}{2} and 92⋅32=274\frac{9}{2} \cdot \frac{3}{2} = \frac{27}{4}.
  • If the ratios of neighbouring terms differ, the sequence is not geometric and has no common ratio.
  • A ratio cannot be formed when a term is 0.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Then the sequence is not geometric. The steps show each ratio, so you can see where they stop agreeing. It may be arithmetic instead; compare the differences.

Yes. A negative ratio makes the signs alternate, and a ratio between −1 and 1 makes the terms shrink toward zero.

Yes. Give both terms and their positions. The steps write the gap as a power of r and solve for it.

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