Section 9 - 4 & 9 - 5: Series In a Nutshell

Section 9 - 4 & 9 - 5: Series In a Nutshell

A series is simply a sum of the values in a sequence that is defined.

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.

Finite Sequences

• Have a first term and a last term (a start and an end)

• If the sequence is finite it will have a sum. The sum is called the series.

• If the sequence is infinite and GEOMETRIC it can have a series IF it converges. To converge means that the common ratio, "r," is less than one.

• If a GEOMETRIC sequence has a common ratio, "r," that is greater than one, we say the GEOMETRIC SERIES diverges.

Fill in the Blanks

ARITHMETIC SERIES: (confirming our understanding) To find an arithmetic series we use the formula

 $S_n=\frac{n}{2}\left(a_1+a_n\right)$  where n is the number of defined terms in the sequence, a1 is the value of the first term and a n is the value of the last term. Find the value of 3 + 10 + 17 + 24 + 31 + 38 using the formula. (Verify your solution by adding the terms.)

Fill in the Blanks

Find the sum given the series 5 + 12 + 16 + 23 + 30 using the formula. You can use your calculator to verify your solution. How many terms? (n/2) Value of the first term? Value of the last term? Sum of those?

Multiple Choice

Given the series 4 + 9 + 14 + 19 + ... + 99; what term number is 99?

• 19

• 20

• 21

Fill in the Blanks

Find the sum of the series 4 + 9 + 14 + 19 + ... + 99 given that 99 is the 20th term.

Multiple Choice

What's the difference? A company pays $10,000 bonus to sales people at the end of their first 50 weeks if they make 10 sales in their first week and improve their sales numbers by 2 each week thereafter. One salesperson qualified for the bonus, with the minimum possible number of sales. How many sales did the salesperson make in week 50?

• 2

• 50

• 100

• 108

Fill in the Blanks

How does this differ? A company pays $10,000 bonus to sales people at the end of their first 50 weeks if they make 10 sales in their first week and improve their sales numbers by 2 each week thereafter. One salesperson qualified for the bonus, with the minimum possible number of sales. How many total sales during the entire 50 weeks?

Fill in the Blanks

GEOMETRIC SERIES: (confirming our understanding) 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 your solution simply adding the terms.)

Multiple Choice

Which of the following is the sum of the series 100 + 20 + 4 + 4/5 + 4/25 + 4/125? What is the first term? What is the common ratio? Raised to the power of?

• 100

• 124.992

• 128

• 390600

Multiple Choice

Given the series 7 + 14 + 28 + ... + 896; 896 is which term in the series?

• 5

• 7

• 6

• 8

Fill in the Blanks

Find the sum of the series 7 + 14 + 28 + ... + 896.

Fill in the Blanks

What's the difference? Given that a ball is dropped from a height of 80 feet and bounces to 80% of its previous height. How high is the ball after 4 bounces?

Multiple Choice

How many bounces until the ball stops bouncing? (AKA the height is equal to zero.)

• 100

• 1000

• 100,000

• 1,000,000

• None of these...it will never stop bouncing

Infinite GEOMETRIC Series

• Will have a sum if the series CONVERGES (has an r < 1)

• Will NOT have a sum if the series DIVERGES (has an r > 1)

Multiple Choice

Which of the following series converges?

• 3 + 6 + 18 + 54 + ...

• 60 + 30 + 15 + 7.5 + ...

• 10 + 30 + 90 + 270 + ...

• .5 + 2 +8 +...

Fill in the Blanks

If an infinite GEOMETRIC series converges, then it can have a sum. The sum is found by using the formula

 $S_{n\ }=\ \frac{a_1}{1-r}$  . Find the sum of the series 60 + 30 + 15 + 7.5 + ...

Multiple Choice

Back to our bouncing ball...If it is dropped from a height of 100 feet, and bounces back to 80% of its previous height, what is the total distance the ball travels in a vertical direction?

• 500 feet

• 1000 feet

• 125 feet

• 250 feet

But wait...there is always an easier way.

 $\sum_{n=1}^nFormula$  

With any series, ARITHMETIC or GEOMETRIC, if you know the explicit formula you can use a special notation, and your calculator, to find the sum of a series. This is using what we call SIGMA notation. You can find the sum of the series using sigma notation. (Math operation on your calculator. For most systems it is the "0" option.)

Fill in the Blanks

 $\sum_{n=1}^{12}3x-10$  

Find the sum of the series given

Fill in the Blanks

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

Find the sum of the series given

Multiple Choice

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

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

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

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

• $\sum_{n=1}^62\left(5\right)^{\left(n\right)}$

Multiple Choice

Use summation notation to find the sum of the series given 109 + 106 + 103 + ... + 73.

• $\sum_{n=1}^{12}3n\ -\ 112;\ -1110$

• $\sum_{n=1}^{13}-3n+112;\ 1183$

• $\sum_{n=1}^{13}-3n\ -112;\ -1729$