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11.4 Series and their notations  (Page 3/18)

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A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?

$2,025

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Using the formula for geometric series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series . Recall that a geometric sequence    is a sequence in which the ratio of any two consecutive terms is the common ratio    ,   r .   We can write the sum of the first n terms of a geometric series as

S n = a 1 + r a 1 + r 2 a 1 + ... + r n – 1 a 1 .

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first   n   terms of a geometric series. We will begin by multiplying both sides of the equation by   r .  

r S n = r a 1 + r 2 a 1 + r 3 a 1 + ... + r n a 1

Next, we subtract this equation from the original equation.

     S n = a 1 + r a 1 + r 2 a 1 + ... + r n – 1 a 1 − r S n = − ( r a 1 + r 2 a 1 + r 3 a 1 + ... + r n a 1 ) ( 1 − r ) S n = a 1 − r n a 1

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for S n , divide both sides by ( 1 − r ) .

S n = a 1 ( 1 − r n ) 1 − r  r ≠ 1

Formula for the sum of the first n Terms of a geometric series

A geometric series    is the sum of the terms in a geometric sequence. The formula for the sum of the first   n   terms of a geometric sequence is represented as

S n = a 1 ( 1 − r n ) 1 − r  r ≠ 1
Given a geometric series, find the sum of the first n terms.
  1. Identify   a 1 ,   r ,   and   n .
  2. Substitute values for   a 1 ,   r , and n into the formula S n = a 1 ( 1 – r n ) 1 – r .
  3. Simplify to find S n .

Finding the first n Terms of a geometric series

Use the formula to find the indicated partial sum of each geometric series.

  1. S 11 for the series  8 + -4 + 2 +  …
  2. ∑ ​ 6 k = 1 3 ⋅ 2 k

  1. a 1 = 8 , and we are given that n = 11.

    We can find r by dividing the second term of the series by the first.

    r = − 4 8 = − 1 2

    Substitute values for a 1 ,   r ,   and   n into the formula and simplify.

    S n = a 1 ( 1 − r n ) 1 − r S 11 = 8 ( 1 − ( − 1 2 ) 11 ) 1 − ( − 1 2 ) ≈ 5.336

  2. Find a 1 by substituting k = 1 into the given explicit formula.

    a 1 = 3 ⋅ 2 1 = 6

    We can see from the given explicit formula that r = 2. The upper limit of summation is 6, so n = 6.

    Substitute values for a 1 ,   r , and n into the formula, and simplify.

    S n = a 1 ( 1 − r n ) 1 − r S 6 = 6 ( 1 − 2 6 ) 1 − 2 = 378
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Use the formula to find the indicated partial sum of each geometric series.

S 20 for the series  1,000 + 500 + 250 +  …

≈ 2 , 000.00

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∑ k = 1 8 3 k

9,840

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Solving an application problem with a geometric series

At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.

The problem can be represented by a geometric series with a 1 = 26 , 750 ;   n = 5 ;   and   r = 1.016. Substitute values for   a 1 ,   r , and n into the formula and simplify to find the total amount earned at the end of 5 years.

S n = a 1 ( 1 − r n ) 1 − r S 5 = 26 , 750 ( 1 − 1.016 5 ) 1 − 1.016 ≈ 138 , 099.03

He will have earned a total of $138,099.03 by the end of 5 years.

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At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

$275,513.31

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Using the formula for the sum of an infinite geometric series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n terms. An infinite series    is the sum of the terms of an infinite sequence. An example of an infinite series is 2 + 4 + 6 + 8 + ...

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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