1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
timofeeve [1]
3 years ago
7

Find the sum or difference. a. -121 2 + 41 2 b. -0.35 - (-0.25)

Mathematics
1 answer:
s344n2d4d5 [400]3 years ago
8 0

Answer:

2

Step-by-step explanation:

The reason an infinite sum like 1 + 1/2 + 1/4 + · · · can have a definite value is that one is really looking at the sequence of numbers

1

1 + 1/2 = 3/2

1 + 1/2 + 1/4 = 7/4

1 + 1/2 + 1/4 + 1/8 = 15/8

etc.,

and this sequence of numbers (1, 3/2, 7/4, 15/8, . . . ) is converging to a limit. It is this limit which we call the "value" of the infinite sum.

How do we find this value?

If we assume it exists and just want to find what it is, let's call it S. Now

S = 1 + 1/2 + 1/4 + 1/8 + · · ·

so, if we multiply it by 1/2, we get

(1/2) S = 1/2 + 1/4 + 1/8 + 1/16 + · · ·

Now, if we subtract the second equation from the first, the 1/2, 1/4, 1/8, etc. all cancel, and we get S - (1/2)S = 1 which means S/2 = 1 and so S = 2.

This same technique can be used to find the sum of any "geometric series", that it, a series where each term is some number r times the previous term. If the first term is a, then the series is

S = a + a r + a r^2 + a r^3 + · · ·

so, multiplying both sides by r,

r S = a r + a r^2 + a r^3 + a r^4 + · · ·

and, subtracting the second equation from the first, you get S - r S = a which you can solve to get S = a/(1-r). Your example was the case a = 1, r = 1/2.

In using this technique, we have assumed that the infinite sum exists, then found the value. But we can also use it to tell whether the sum exists or not: if you look at the finite sum

S = a + a r + a r^2 + a r^3 + · · · + a r^n

then multiply by r to get

rS = a r + a r^2 + a r^3 + a r^4 + · · · + a r^(n+1)

and subtract the second from the first, the terms a r, a r^2, . . . , a r^n all cancel and you are left with S - r S = a - a r^(n+1), so

(IMAGE)

As long as |r| < 1, the term r^(n+1) will go to zero as n goes to infinity, so the finite sum S will approach a / (1-r) as n goes to infinity. Thus the value of the infinite sum is a / (1-r), and this also proves that the infinite sum exists, as long as |r| < 1.

In your example, the finite sums were

1 = 2 - 1/1

3/2 = 2 - 1/2

7/4 = 2 - 1/4

15/8 = 2 - 1/8

and so on; the nth finite sum is 2 - 1/2^n. This converges to 2 as n goes to infinity, so 2 is the value of the infinite sum.

You might be interested in
12. A drawer contains 2 red socks, 6 white socks,
777dan777 [17]

Answer:

Probability that first sock is blue is 0.33 and Probability that second sock is red is 0.16

Step-by-step explanation:

Number of red socks = 2

Number of white socks = 6

Number of blue socks = 4

Total socks in drawer = 2+6+4 = 12

The formula used to calculate probability is: Probability=\frac{Number \ of \ favourable \ outcomes}{Total \ number \ of  \ outcomes}

We are given you draw out a  sock, return it, and draw out a second sock.

We need to find the probability that the first sock is blue and the  second sock is red?

Using formula:

Probability that first sock is blue = 4/12 = 1/3 = 0.33

Probability that second sock is red = 2/12 = 1/6 = 0.16

So, Probability that first sock is blue is 0.33 and Probability that second sock is red is 0.16

3 0
2 years ago
Tania wanted to watch the movie Earwig and the witch. On HBO you have to subscribe for $14.99 to be able to watch it. She has a
IrinaK [193]

Answer:

aprox 33%

Step-by-step explanation:

6 0
3 years ago
Someone please helpp me quick awarding brainliest
fredd [130]

Answer:

A

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Find the point on the line and the lines slope y+1=-3/2(x+7)
klemol [59]

Answer:

A point on the line would be (-3, -7) and the slope if -3/2

Step-by-step explanation:

We can find a point on the line by choosing any value for x and putting it in to the equation. Then we can find the y value. For the purpose of this problem, we will use x = -3.

y + 1 = -3/2(x + 7)

y + 1 = -3/2(-3 + 7)

y + 1 = -3/2(4)

y + 1 = -6

y = -7

And we can find the slope simply by looking for the number outside of the parenthesis on the right side of the equal sign (-3/2)

8 0
3 years ago
Three men working together can finish a house repair job in 10 days. How manydays will 5 men working at the same rate finish the
Vanyuwa [196]

Answer:

17 days

Step-by-step explanation:

Number of days in which 3 men can finish a house repair job = 10 days

Number of days in which 1 man can finish a house repair job = \frac{10}{3} days

Therefore,

Number of days in which 5 men can finish a house repair job = Number of days in which 1 man can finish a house repair job × 5

=\frac{10}{3}(5)=\frac{50}{3}=16.6 ≈ 17 days

5 0
2 years ago
Other questions:
  • What is the slope of a line that is perpendicular to the line y = -1/2 x + 5?
    15·1 answer
  • You want to deliver a package to japan.the fed ex delivery service charges a $15 flat fee plus $2 per additional kilogram. You c
    8·1 answer
  • What is 0.08% written as a decimal?
    13·1 answer
  • Create a real life example using either temperature or money to compare -3 and -5
    5·2 answers
  • Use the identity (x2+y2)2=(x2−y2)2+(2xy)2 to determine the sum of the squares of two numbers if the difference of the squares of
    12·2 answers
  • a hose fills a tank at rate of 30 gallons per minute. what is rate in liters per hour. 1 gallon=3.79 liters
    14·1 answer
  • Explain how to multiply the following whole numbers 21 x 14
    8·1 answer
  • A major credit card company requires monthly payments equal to or greater than 8% of the total balance.A consumer has a credit c
    11·1 answer
  • Why is the square root of 7 irrational
    13·1 answer
  • KE= 0.5 .m.v2 OR PE= m.g.h Kinetic and Potential Energy Practice Problems. Solve the following problems and show your work! 1. A
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!