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.

Multiple Choice

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

• 19

• 20

• 21

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

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

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 +...

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.)

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.

• $\sum_{n=1}^{12}3n\ -\ 112;\ -1110$

• $\sum_{n=1}^{13}-3n+112;\ 1183$

• $\sum_{n=1}^{13}-3n\ -112;\ -1729$