### Posts Tagged ‘sequences’

This week we learned about geometric sequences and series. A Geometric sequence differs from an arithmetic sequence because instead of adding a common difference, a common ratio is multiplied by the term to get the next term.

the equation used to find a term in a sequence is $t_n=ar^{n-1}$

in this equation $t_1$ is represented by $a$

for example: 5,10,20,40… find $t_{10}$ $t_{10}=5(2)^{10-1}$ $t_{10}=5(2)^9$ $t_{10}=5(512)$ $t_{10}=2560$

A finite geometric series can be found using the equation 4, 9, 14, 19, 24 $d=5$, $t_1=4$ $t_n=t_1+(n-1)d$ $t_n=4+(n-1)5$ $t_n=4+5n-5$ $t_n=5n-1$

using this information, we can find any value of $t$ using the equation $t_n=5n-1$. We can plug in any term into $n$ to find $t_n$

this equation will work for any term within the sequence, as long as the starting term and common difference are constant. If either of these change it is no longer an arithmetic sequence.

To find $S_{50}$, the sum of all 50 terms, we must use the equation $S_n={n}{2}(a+1)$

In this equation, $n=$ number of terms(50) and $a=$ the first term(4) $S_{50}={50}{2}(4+1)$ $S_{50}=25(5)$ $S_{50}=125$