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
Arisa [49]
3 years ago
13

If the sum of the even integers between 1 and k, inclusive, is equal to 2k, what is the value of k?

Mathematics
1 answer:
Alisiya [41]3 years ago
5 0
If k is odd, then

\displaystyle\sum_{n=1}^{\lfloor k/2\rfloor}2n=2\dfrac{\left\lfloor\frac k2\right\rfloor\left(\left\lfloor\frac k2\right\rfloor+1\right)}2=\left\lfloor\dfrac k2\right\rfloor^2+\left\lfloor\dfrac k2\right\rfloor

while if k is even, then the sum would be

\displaystyle\sum_{n=1}^{k/2}2n=2\dfrac{\frac k2\left(\frac k2+1\right)}2=\dfrac{k^2+2k}4

The latter case is easier to solve:

\dfrac{k^2+2k}4=2k\implies k^2-6k=k(k-6)=0

which means k=6.

In the odd case, instead of considering the above equation we can consider the partial sums. If k is odd, then the sum of the even integers between 1 and k would be

S=2+4+6+\cdots+(k-5)+(k-3)+(k-1)

Now consider the partial sum up to the second-to-last term,

S^*=2+4+6+\cdots+(k-5)+(k-3)

Subtracting this from the previous partial sum, we have

S-S^*=k-1

We're given that the sums must add to 2k, which means

S=2k
S^*=2(k-2)

But taking the differences now yields

S-S^*=2k-2(k-2)=4

and there is only one k for which k-1=4; namely, k=5. However, the sum of the even integers between 1 and 5 is 2+4=6, whereas 2k=10\neq6. So there are no solutions to this over the odd integers.
You might be interested in
HEEEEEEEEEEEEEEEEEEEEEEEELPPPPPPPPPPPPPP
Rasek [7]
Answer:
slope = -9/2

Explanation:
The form of the slope-intercept equation is:
y = mx + c
where:
m is the slope
c is the y-intercept

The given is:
y = -9/2 x

Comparing the given with the general form, we will find that:
m is the slope = -9/2
c is the y-intercept = 0

Hope this helps :)
y = -9/2

7 0
3 years ago
Read 2 more answers
Where do the lines y=2x+1 and y=-5x-6 intersect?
kipiarov [429]
Answer: (-1,-1)

Explanation:

I added a picture
3 0
3 years ago
Read 2 more answers
The difference of the squares of two positive consecutive even integers is 84. Find the integers.
ziro4ka [17]

Answer:

The two integers are 20 and 22.

Step-by-step explanation:

Let x be the first integer.

(x+2)^{2}-x^{2} = 84\\x^{2} +4x+4-x^{2} = 84\\4x = 80\\x = 20

3 0
3 years ago
Alan deposited $300 into a savinga account. At the end of the first year, the amount of money in the account has increased to $3
Sergeeva-Olga [200]
The Percent increase is 12%
8 0
3 years ago
The cost of 6 sandwiches and 4 drinks is $53. The cost of 4 sandwiches and 6 drinks is $47. How much does one sandwich cost?
melomori [17]

In this question, we're trying to find cost of one sandwich.

To find this, we must make a systems of equation from the given information:

We would represent sandwiches as "s" and drinks as "d".

Systems of equations:

6s + 4d = 53

4s + 6d = 47

Solve for s:

6(6s + 4d = 53)

-4(4s + 6d = 47)

36s + 24d = 318

-16s - 24d = -188

--------------------------

20s = 130

Divide both sides by s

s = 6.50

This means that one sandwich costs $6.50

Answer:

$6.50

8 0
3 years ago
Other questions:
  • A gardener is fencing off a rectangular area with a fixed perimeter of 72 ft.What is the maximum area?
    10·1 answer
  • PLEASE HELP ME IM SO CONFUSED The volume of a cube is x³ cubic units. The surface area of the cube is x³ square units. What is t
    12·1 answer
  • 1.)12.5+(-31.5)=<br>2.)-31.91+(-16.7)=<br>3.)1/6+(-11/12)+2/3=<br>4.)29+(-15.7)+(-31.05)=
    14·1 answer
  • Plss help 0-0
    13·1 answer
  • F/4 + 16 &lt; 49<br> Solve the inequality
    9·1 answer
  • The coordinate grid shows three points P, Q, and R:
    11·2 answers
  • HELP ME PLEASE <br> Again lol
    9·1 answer
  • Please help!!!!<br> A 1 and 3 only<br> B 2 only<br> C 1 and 2 only<br> D 1 only
    6·1 answer
  • Hello, may someone plz help me
    13·1 answer
  • Hello I need help the picture is down below
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!