Geometric and Arithmetic Sequences

The common ratio here is determined by

$$Common\ Ratio\ =\frac{\sec ond\ term}{first\ term}=\frac{45.5}{13}=3.5$$

So each term is multiplied by 3.5 to get the next term in the sequence.

The common ratio here is determined by

$$Common\ Ratio\ =\frac{\sec ond\ term}{first\ term}=\frac{31.5}{52.5}=0.6$$

So each term is multiplied by 0.6 to get the next term in the sequence.

The common difference here is determined by

$$Common\ Difference\ =\sec ond\ term-first\ term$$ $$=18-13=5$$

So each term has 5 added to it to get the next term.

The common difference here is determined by

$$Common\ Difference\ =\sec ond\ term-first\ term$$ $$=37.5-52.5=15$$

So each term has 15 subtracted from it to get the next term.

Remember to find the common difference of an arithmetic sequence subtract the first and second term like so:

$$10-5=5$$

To Find the common difference subtract 8-1 and you get the common difference of 7, so you add 7 to each term to get the next and you get 15+7=22, 22+7=29, 29+7=36 so the next 3 in the sequence are:

22, 29, and 36

Remember to find the common ratio you must divide the second term by the first term:

$$3,\ 6,\ 12,\ 24,\ 48,\ .\ .\ .\ \rightarrow\ common\ ratio\ =\frac{6}{3}=2$$

$$2,\ 8,\ 32,\ 128,\ .\ .\ .\ \rightarrow\ common\ ratio\ =\frac{8}{2}=4$$

$$144,\ 36,\ 9,\ 2.25,\ .\ .\ .\ \rightarrow\ common\ ratio\ =\frac{36}{144}=0.25$$

If the explicit formula for a geometric sequence is written: $$a_n=a_1\cdot\left(r\right)^{n-1}$$

Where a₁ = the first term and r = the common ratio we have that a₁ =3 and $$r=\frac{6}{3}=2$$

We get:

$$a_n=3\cdot\left(2\right)^{n-1}$$

If the explicit formula for a geometric sequence is written: $$a_n=a_1\cdot\left(r\right)^{n-1}$$

Where a₁ = the first term and r = the common ratio we have that a₁ =144 and $$r=\frac{36}{144}=0.25$$

We get:

$$a_n=144\cdot\left(0.25\right)^{n-1}$$