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
tekilochka [14]
2 years ago
12

Use the approach in Gauss's Problem to find the following sums of arithmetic

Mathematics
1 answer:
Agata [3.3K]2 years ago
3 0

a. Let S be the first sum,

S = 1 + 2 + 3 + … + 97 + 98 + 99

If we reverse the order of terms, the value of the sum is unchanged:

S = 99 + 98 + 97 + … + 3 + 2 + 1

If we add up the terms in both version of S in the same positions, we end up adding 99 copies of quantities that sum to 100 :

S + S = (1 + 99) + (2 + 98) + … + (98 + 2) + (99 + 1)

2S = 100 + 100 + … + 100 + 100

2S = 99 × 100

S = (99 × 100)/2

Then S has a value of

S = 99 × 50

S = 4950

Aside: Suppose we had n terms in the sum, where n is some arbitrary positive integer. Call this sum ∑(n) (capital sigma). If ∑ is a sum of n terms, and we do the same manipulation as above, we would end up with

2 ∑(n) = n × (n + 1)   ⇒   ∑(n) = n (n + 1)/2

b. Let S' be the second sum. It looks a lot like S, but the even numbers are missing. Let's put them back, but also include their negatives so the value of S' is unchanged. In doing so, we have

S' = 1 + 3 + 5 + … + 1001

S' = (1 + 2 + 3 + 4 + 5 + … + 1000 + 1001) - (2 + 4 + … + 1000)

The first group of terms is exactly the sum ∑(1001). Each term in the second grouped sum has a common factor of 2, which we can pull out to get

2 (1 + 2 + … + 500)

so this other group is also a function of ∑(500), and so

S' = ∑(10001) - 2 ∑(500) = 251,001

However, we want to use Gauss' method. We have a sum of the first 501 odd integers. (How do we know there 501? Starting with k = 1, any odd integer can be written as 2k - 1. Solve for k such that 2k - 1 = 1001.)

S' = 1 + 3 + 5 + … + 997 + 999 + 1001

S' = 1001 + 999 + 997 + … + 5 + 3 + 1

2S' = 501 × 1002

S' = 251,001

c/d. I think I've demonstrated enough of Gauss' approach for you to fill in the blanks yourself. To confirm the values you find, you should have

3 + 6 + 9 + … + 300 = 3 (1 + 2 + 3 + … + 100) = 3 ∑(100) = 15,150

and

4 + 8 + 12 + … + 400 = 4 (1 + 2 + 3 + … + 100) = 4 ∑(100) = 20,200

You might be interested in
If someone can pls give the answer with steps that would be greatly appreciated :)
AleksAgata [21]

Answer:

Step-by-step explanation:

Let C be the number of correct answers

Let B be the number of questions not answered

Let W be the wrong answers

Total = 2*C + B -  W

It's a trinomial. There are three unrelated terms.

5 0
3 years ago
(−7, 5) and (7, 5) on the coordinate grid below show the positions of two players of a football team: A coordinate plane is show
galina1969 [7]

Answer:

Player 1's position is Player 2's position reflected across the y-axis; only the signs of the x-coordinates of Player 1 and Player 2 are different.

Step-by-step explanation:

When you reflect a point (x, y) across the y-axis, the y-coordinate remains the same, but the x-coordinate gets the opposite sign: it becomes (-x, y).

Thus, if a point P, say, (7,5) is reflected across the y-axis, its reflection P' becomes(-7,5)

3 0
3 years ago
How do the unit rates of miles per gallon compare for these two vehicles?
Bad White [126]
The van travels 5 more miles per gallon than the van
5 0
2 years ago
????!!!!??!!?!!?!!?!!?!!!!?!!!!
larisa [96]

Answer:

<h3>A.</h3>

Step-by-step explanation:

if both increases the volume also increase as the same i.e.

twice length and twice radius = twice the volume of wax

<h2 /><h2><u>MARK ME AS BRAINLIST</u> </h2>
7 0
2 years ago
Can someone pls help me pls
Norma-Jean [14]

Answer:

  a.

Step-by-step explanation:

Usually, we represent losses with negative numbers. Using that representation, a loss of $975 would be represented by the integer -975.

__

The additive inverse of this number is also called its "opposite." It is found by changing the sign: additive inverse = +975.

__

The sum of a number and its additive inverse is zero. The opposite of a loss is a gain. The value +975 represents the gain (the investment's earnings) that would bring the net change in investment value back to zero.

7 0
2 years ago
Other questions:
  • 150 decreased by 60%
    10·1 answer
  • Graph the system of equations on your graph paper to answer the question. {y=−x+3
    13·1 answer
  • How much more is five ninth’s greater than 4 fifteenths
    7·1 answer
  • This is from my Sociology class and there's not that option soooo
    15·1 answer
  • Solve equation 66= -11y
    12·2 answers
  • The first segment she draws is 6 centimeters in length, and the second segment is 8 centimeters long. The measure of the angle b
    15·1 answer
  • A metallurgist wants to make 130 grams of an alloy that is to be 20% copper. She has samples that are 14% copper and 22% copper.
    8·1 answer
  • Please help!! im very confused
    10·2 answers
  • HELP NEEDED!
    6·1 answer
  • Which equations are correct?
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!