February 10, 2018 – Manroop's Blog

This week in Pre-Calc 11 we continued to learn about different series and sequences. This week we learned about Geometric Sequences and Series.

What is a Geometric Sequence? A Geometric Sequence is a group of numbers that is multiplied by a constant ratio.

Ex. 2 ,10 ,50 ,250…

The common ratio for this sequence is $\frac{5}{1}$ or you can represent the common ratio by using the number 5. The common ratio is represented by the letter $r$ . A Geometric Series can NEVER have a dividing pattern. The common ratio of a geometric sequence CANNOT equal to 0 or 1.

Ex. 12, 6, 3, 1.5…

For this pattern $r=\frac{1}{2}$ although at first you might think that the pattern is dividing by 2 each time. In a geometric sequence the numbers have to be MULTIPLIED by a constant ratio each time, so this pattern can be represented as $r=\frac{1}{2}$ because multiplying by $\frac{1}{2}$ is the same as dividing by 2 each time.

How do you find r? The formula to find r is $r=\frac{t_n}{t_(n-1)}$

Ex. $\frac{1}{3},\frac{-1}{6},\frac{1}{12}$

$t_n=\frac{1}{12}$

$t_(n-1)=\frac{-1}{6}$

$r=\frac{t_n}{t_(n-1)}$

$r=\frac{1}{12}\div\frac{-1}{6}$

$r=\frac{1}{12}\times\frac{6}{-1}$

$r=\frac{6}{-12}=\frac{-1}{2}$

$r=\frac{-1}{2}$

How do you find $t_n$ ? The formula for $t_n$ is $t_n=a*r^{n-1}$ . Instead of the first term being represented as $t_1$ , geometric sequences represent the first term as “a”. Similar to arithmetic sequences, geometric sequences also use “n” to represent the number of terms.

Ex. Find $t_{11}$ for the following geometric sequence: 5, 10, 20, 40…

$r=2$

$a=5$

$t_n=a*r^{n-1}$

$t_{11}=5*2^{10}$

$t_{11}=5*1024$

$t_{11}=5120$

What is a Geometric Series? A Geometric Series is similar to a geometric sequence except you replace the commas with a plus sign to find the sum of a certain number of terms.

Ex. $2+4+8+16$