Arithmetic and Geometric Series

Arithmetic and Geometric Series

A series is a sum of the values in a sequence.
An infinite series goes on forever,  $\infty$  or ...

A finite series is the sum of a specific number of terms.

Multiple Choice

Which of the following sequences is finite?

• 2, 4, 6, 8, ...

• 3, 7, 11, 15, 19

• 10, -20, 40, -80, ...

• All of the sequences are finite.

Fill in the Blanks

To find the sum of an ARITHMETIC series we use the formula $S_n\ =\ \frac{n}{2}\left(a_1+a_n\right)$  where n is the number of terms in the sequence, a_1 is the first term and a_n is the last term.

Find the value of  $3+10+17+24+31+38$  using the formula.  Verify the solution by adding the terms.

Multiple Choice

Given the series $4+9+14+19+...+99$  

what term number (n) is 99?

Hint - use  $a_n=a_1+\left(n-1\right)d$  

• 19

• 21

• 20

• 22

Fill in the Blanks

 $4+9+14+19+...+99$  

Find the sum of the series given that 99 is the 20th term.

 $S_n=\frac{n}{2}\left(a_1+a_n\right)$  

Fill in the Blanks

To find a GEOMETRIC series, you must use the formula $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$  

Find the sum of the series  $2+4+8+16+32+64$  You can verify the solution by adding the terms.

Multiple Choice

What is the sum of the series $100+20+4+\frac{4}{5}+\frac{4}{25}+\frac{4}{125}$ ? Hint - Find the first term. Find the common ratio.  Raised to what power?

 $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$  

• 100

• 128

• 124.992

• 390.600

Multiple Choice

Given the series  $7+14+28+...+896$  What term is 896?

Use  $a_n=a_1\cdot r^{\left(n-1\right)}$  

• 5

• 6

• 7

• 8

Fill in the Blanks

What is the sum of $7+14+28+...+896$ ? 

 $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$   

Sigma Notation

 $\sum_{n=1}^nExplicit\ Formula\ \left(S_n\ =\ \right)$  
With any series (arithmetic or geometric) you can use Sigma Notation to find the sum.  If you don't have a calculator that has Sigma notation, you can use the explicit formula to find a_1, a_n, and n.  Then use S_n formula to find the sum.

Fill in the Blanks

 $\sum_{n=1}^{12}3x-10$  

Find the sum of the arithmetic series.

HINT - find a_1, a_n, and n.  Then use  $S_n=\frac{n}{2}\left(a_1+a_n\right)$  

Multiple Choice

Which is the correct summation notation for $5+10+20+40+80+160+320$  ?

•  $\sum_{n=1}^75\left(2\right)^{\left(n-1\right)}$  

•  $\sum_{n=1}^8\left(5\left(2\right)^{\left(n-1\right)}\right)$  

•  $\sum_{n=1}^72\left(5\right)^{\left(n-1\right)}$  

•  $\sum_{n=1}^62\left(5\right)^n$  

Fill in the Blanks

 $\sum_{n=1}^82\left(3\right)^{\left(n-1\right)}$  

Find the sum of the series