Find the common ratio and the next term of a geometric sequence

For consecutive terms 14 and 35, which calculation finds the common ratio?

Change R through all four settings. Read the next two terms and compare the separate points. The first term and the graph scale stay fixed

With a positive starting term, what happens when R is 0.5?

One multiplier at every step

The list 5, 15, 45, 135 multiplies by 3 each time. That fixed multiplier is called the common ratio. The amount added does not stay the same

1. 1Forward step Multiply the current term by 3. \(\displaystyle 5\times3=15\)
2. 2Recover the multiplier Divide the later term by the previous term \(\displaystyle 15\div5=3\)

Consecutive terms

\(\displaystyle 7,\ 21,\ 63\)

Find the common ratio. Give a whole number

Check later divided by earlier. Enter a whole-number numeral, not a fraction or decimal, even if it has the same value

A ratio below one is still positive

For 48, 36, 27, each term is three quarters of the previous term. The losses are 12 then 9, so this is not a constant subtraction

1. 1Check both quotients Both simplify to three quarters \(\displaystyle \frac{36}{48}=\frac{27}{36}=\frac34\)
2. 2State the ratio, not the loss The positive ratio is three quarters. The percentage decrease is a different quantity

Consecutive terms

\(\displaystyle 40,\ 30,\ 22.5\)

Write the common ratio as a decimal

Check the quotient, then write it in decimal notation. An equivalent fraction has the same value but is not the requested form

The displayed terms are 9, 18, 36, 73. Do all the consecutive pairs have the same ratio?

The special case of a ratio of one

The sequence 11, 11, 11, 11 multiplies by 1 each step. It also adds 0 each step, so this nonzero constant sequence is both geometric and arithmetic. It is neither increasing nor decreasing

1. 1What can a finite list establish? It shows whether the displayed steps fit a rule. It cannot prove how an arbitrary unseen continuation will behave
2. 2Do not divide by zero The quotient test needs a nonzero previous term. Our ratio calculations here use positive terms

Work backwards with division

Suppose the ratio is three halves and one term is 54. To recover the previous term, undo multiplication by three halves

1. 1Divide by the ratio Dividing by three halves is multiplying by two thirds \(\displaystyle 54\div\frac32=54\times\frac23=36\)
2. 2Check forwards The recovered term should give the known term after one multiplication \(\displaystyle 36\times\frac32=54\)

Common ratio

\(\displaystyle \frac25\)

Known term

\(\displaystyle 28\)

Find the term immediately before 28. Give its exact value

Undo one multiplication by dividing the known term by the ratio

A sequence starts with a positive value and multiplies by the same positive number less than 1 at every step. Which description follows?