How many terms of the series of - 3+0+3+6+9+...are needed to give a sum of 105?​

Answers

Answer 1

Answer:

10

Step-by-step explanation:

Remember that the formula for the sum of an arithmetic series is:

[tex]S=\frac{k}{2}(a+x_k)[/tex]

Where k is the number of terms, a is the initial term, and x_k is the last term of the series.

We essentially want to find k, the number of terms, given that the sum S is equal to 105. So, substitute 105 into our equation:

[tex]105=\frac{k}{2}(a+x_k)[/tex]

To do so, we need to final term x_k. We don't know what it is yet, but that doesn't matter. All we need to do is to write it in terms of k. First, remember that the standard form for the explicit formula of an arithmetic sequence is:

[tex]x_n=a+d(n-1)[/tex]

Where a is the first term, d is the common difference, and n is the nth term.

From our sequence, we can see that the first term is -3.

Also, we can determine that our common difference is +3, since each subsequent term is 3 more than the previous one. -3+3 is 0, 0+3 is 3, 3+3 is 6, and so on.

Therefore, our explicit formula is:

[tex]x_n=-3+3(n-1)[/tex]

Therefore, our final term, x_k, will be if we substitute k for n. So, we can acquire the equation:

[tex]x_k=-3+3(k-1)[/tex]

Now that we know what x_k is, we can substitute that into our original equation:

[tex]105=\frac{k}{2}(a+x_k)[/tex]

Substitute the equation into x_k. Also, let's substitute -3 (our first term) for a. So:

[tex]105=\frac{k}{2}(-3+(-3+3(k-1)))[/tex]

And now, all we have to do is to solve for k.

First, distribute the 3:

[tex]105=\frac{k}{2}(-3+(-3+3k-3))[/tex]

Add within the parentheses:

[tex]105=\frac{k}{2}(3k-9)[/tex]

Multiply both sides by 2. This removes the fraction on the right:

[tex]210=k(3k-9)[/tex]

Distribute. We will get a quadratic:

[tex]210=3k^2-9k[/tex]

So, let's solve for k. Let's divide everything by 3:

[tex]70=k^2-3k[/tex]

Subtract 70 from both sides:

[tex]0=k^2-3k-70[/tex]

Factor. We can use -10 and 7. So:

[tex]0=(k-10)(k+7)[/tex]

Zero Product Property:

[tex]k-10=0\text{ or } k+7=0[/tex]

Solve for k for each equation:

[tex]k=10\text{ or } k=-7[/tex]

-7 doesn't make sense (we can't have -7 terms). Remove that solution. So, we are left with:

[tex]k=10[/tex]

Therefore, the number of terms we have in our series for our sum to be 105 is 10.

And we're done!


Related Questions

Where in real life do you work with radical expressions and functions? Give an example of your own.

Answers

Engineering to make things

The sum of three numbers is 82 The second number is 3 times the third. The third number is 8 more than the first. What are the numbers?

Answers

Answer:

22 46 15

Step-by-step explanation:

i looked it up because i was even confused. but i think this is right. hope this helps.

Find the percent change. State whether it is an increase or decrease. Round to the nearest tenth of a percent. From 50 to 42

Answers

Answer:

16% decrease

Step-by-step explanation:

-2X=-42
What do I do to this problem

Answers

You divide -42 by -2 and then you get X
X= 21

if 49+ b2= 169, what is b2?

Answers

b=60 ..

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Practice isolating the variable to solve equations with
variables on both sides.
Assignment
Consider the equation 9 - 4x = 2x.
Which equation resulted from adding 4x to both sides to isolate the variable term?
O9-2x = 4x
O 9 = -2x
O 9 =6x
O 4x=2x-9

Answers

Answer:

9 =6x

Step-by-step explanation:

We know this is the answer because the -4x was cancelled out left to the equal sign, and 2x has now become 6x after adding 4x.

Answer:

9/6 or 1.5

Step-by-step explanation:

both are right

Answer the question in the file

Answers

Answer:

30

Step-by-step explanation:

50-2×(3^2+1)

50-2×(9+1)

50-2×(10)

50-20

30

Hey there!

ANSWER: [tex]30[/tex]EXPLANATION:

[tex]50-2*(3^2+1)=?[/tex]

You will need to use the order of operations to solve. We will need to start with parenthesis. The parenthesis includes exponents which comes after.

[tex](3^2+1)=10[/tex]

Now go to multiplication.

[tex]2*10=20[/tex]

Now move to subtraction since that's what we have left.

[tex]50-20=30\\30(ANSWER)[/tex]

Hope this helps!

[tex]\text {-TestedHyperr}[/tex]

The double number line shows that
4
44 pounds of almonds cost
$
34
$34dollar sign, 34.

Answers

Answer:

round them up

Step-by-step explanation:

(7x+40)+(6x+43)=180 x? Z?

Answers

[tex](7x + 40) + (6x + 43) = 180\\7x + 40 + 6x + 43 = 180\\7x + 6x = 180 - 40 - 43\\13x = 97\\x = \frac{97}{13}[/tex]

I'm not sure what you mean by Z but x is [tex]\frac{97}{13}[/tex]

Question 3
1 pts
Which number could be used as a counterexample to show that the conjecture
"All multiples of 3 are odd" is false?
12
027
O 33
O 15

Answers

The correct answer is A. 12

Explanation:

In mathematics, odd numbers are those numbers that cannot be divided into 2 without using a decimal number, for example, 13 ÷ 2 = 2.5 (decimal number). This does not occur with even numbers such as 26 because these can be divided into 2 and the result is an integer rather than a decimal (26 ÷ 2 =13).

On the other hand, a counterexample is a situation that shows an exception to the rule. In the rule "All multiples of 3 are odd", one counterexample is the number 12 because this is an even number (12 ÷2 = 6). Moreover, this number is a multiply of 3 as 3 can be multiplied by 4 to obtain 12, which shows multiplies of 3 can be even numbers.

One sample has a mean of M=7.2 and a second sample has a mean of M=12.8. The two sample are combined into a single set of scores. a. What is the mean for the combined set if both the samples have n=5? b. What is the mean for the combined set if the first sample has n=5 and the second sample had n=35?

Answers

Answer:

a. M = 50

b. M = 242

Step-by-step explanation:

mean =  [tex]\frac{sum of scores}{number of samples}[/tex]

⇒ sum of scores = mean x number of samples

Let the mean be represented by M, and the number of samples be represented by n,

sum of scores = M x n

a. When n = 5 for both samples,

The first sample's sum of scores = M x n

                                     = 7.2 x 5

                                     = 36

The second sample's sum of scores = M x n

                                     = 12.8 x 5

                                     = 64

The mean for the sum of scores of the combined set = [tex]\frac{36 + 64}{2}[/tex]

                                     = 50

When both samples have n = 5, the mean of the combined set is 50.

b. When the first sample has n = 5, the sum of scores = 36.

If n = 35 for the second sample, the sum of scores = M x n

                                                          = 12.8 x 35

                                                          = 448

The mean for the sum of scores of the combined set = [tex]\frac{36 + 448}{2}[/tex]

                                                    = 242

Seven oxide thickness measurements of wafers are studied to assess quality in a semiconductor manufacturing process. The data (in angstroms) are: 1264, 1288, 1272, 1304, 1280, 1301, and 1270. Calculate (a) the sample average and (b) sample standard deviation. Round both answers to 1 decimal places.

Answers

Answer:

a. Mean = 1,282.7 Angstroms

b. Standard deviation = 15.5

Step-by-step explanation:

a. Firstly, we are told to calculate the sample average. This is same as the mean;

Mathematically that would be;

The value of the sum of terms/ The number of terms.

= (1264 + 1288 + 1272 + 1304 + 1280 + 1301 + 1270)/7 = 8979/7 = 1282.7

b. Standard deviation

Mathematically, this is the square root of variance

= √{(x - mean)^2/N-1}

Please check attachment for the breakdown of this

Solve for t.
-3t + 3 – 7t = -90 – 7

Answers

Answer:

t = 10

Step-by-step explanation:

-3t + 3 - 7t = -90 - 7

Simplify both sides of the equation

That's

- 10t + 3 = - 97

Using the subtraction property subtract 3 from both sides of the equation

We have

- 10t + 3 - 3 = -97 - 3

- 10t = - 100

Divide both sides by - 10

[tex] \frac{ - 10t}{ - 10} = \frac{ - 100}{ - 10} [/tex]

We have the final answer as

t = 10

Hope this helps you

Answer:

Step-by-step explanation:

-3t + 3 - 7t = -90 - 7

-10t + 3 = -97

-10t = -100

t = 10

Which shows only a vertical translation from the shaded figure onto the unshaded figure?
On a coordinate plane, a triangle is shifted up.
On a coordinate plane, a triangle is shifted to the left.
On a coordinate plane, a triangle is enlarged to form another triangle.
On a coordinate plane, a triangle is shifted up and to the left.

Answers

Answer:A

Step-by-step explanation:

Answer: Answer is A

Step-by-step explanation: i got it right on edge 2023 <33

Find the midpoint of the line segment with the given endpoints (-4,4) (5,-1)

Answers

Answer:

(1/2), (3/2)

Step-by-step explanation:

midpoint fomula

(x1+x2)/ 2,  (y1+y2)/2

(5-4)= 1    1/2 = (1/2)

(-1+4)=3    3/2=(3/2)

Answer :

A : (x₁ , y₁) = (-4 , 4)

B : (x₂ , y₂) = (5 , -1)

Formula of [tex]\tt{midpoint}[/tex] is given by

(x , y) = (x₂ + x₁)/2 , (y₂ + y₁)/2

_________________________________

⇒ (x , y) = (x₂ + x₁)/2 , (y₂ + y₁)/2

⇒ (x , y) = [5+(-4)]/2 , [-1+4]/2

⇒ (x , y) = 1/2 , 3/2

⇒ (x , y) = (0.5 , 1.5)

Mateo scored in the 55th percentile of his class on the latest math exam. What does this mean in relation to the data set, and what did he score on the exam?

Answers

Answer:

d. Mateo scored as well as or better than 55% of his classmates, with a score of 82.

Step-by-step explanation:

Mateo scored in the 55th percentile of his class on the latest math exam. What does this mean in relation to the data set and what did he score on the exam?

{35, 42, 55, 68, 69, 72, 72, 77, 78, 78, 82, 85, 87, 88, 90, 93, 94, 96, 96, 99)  

a. Mateo scored a 55% on his exam. b. Mateo scored as well as or better than 55% of his classmates, with a score of 78. c. Mateo scored worse than 55% of his classmates, with a score of 78. d. Mateo scored as well as or better than 55% of his classmates, with a score of 82. e. Mateo scored worse than 55% of his classmates, with a score of 82. Solution

There are scores of 20 students. Each score is equivalent of 100%/20 = 5%

The easy way to understand Mateo' s place is to compare scores against 5% slots.

See attached.

We can also get the average score

Sum

∑ {35, 42, 55, 68, 69, 72, 72, 77, 78, 78, 82, 85, 87, 88, 90, 93, 94, 96, 96, 99) = 1556

Average or mean score:

1556/20 = 77.8

This is matching the 50% score.

As we see Mateo scored 82 which is matching 55%.

Correct option is d.

The center of circle P is at (-3, 8) and the radius is 12
units. Find its equation.

Answers

Answer:

The answer is

x² + y² + 6x - 16y - 71 =0

Step-by-step explanation:

The equation of a circle is given by

( x - a)² + ( y - b)² = r²

where ( a , b) is the center of the circle and

r is the radius

From the question the center is

P (-3, 8) and radius = 12

Substitute the values into the above formula and solve

That's

( x + 3)² + ( y - 8) = 12²

Expand the terms

That's

x² + 6x + 9 + y² - 16y + 64 = 144

Arrange the terms

x² + y² + 6x - 16y + 9 + 64 - 144 = 0

We have the final answer as

x² + y² + 6x - 16y - 71 =0

Hope this helps you

Cost of wood is $9.50. How much would it cost Matt to buy 2 1/2 metres?

Answers

Answer:

23.75

Step-by-step explanation:

how this helped bye I would appreciated if u liked my comment tho

The cost of 2 1/2 meters of wood is $23.75

How to determine the cost of 2 1/2 meters?

The given parameters are:

Unit rate = $9.50 per minutesNumber of meters = 2 1/2 meters

The total cost is then calculated as:

Total = 9.50 * 2 1/2

Evaluate the product

Total = 23.75

Hence, the cost of 2 1/2 meters of wood is $23.75

Read more about rates at:

https://brainly.com/question/843074

#SPJ9

What is 33/8 in a mixed number

Answers

Answer: 4 1/8

Step-by-step explanation: To change an improper fraction to a mixed number, you need to first divide the numerator by the denominator as the fraction sign is essentially just a division sign. In this case, you do 33 divided by 8, which would get you 4 with a remainder of 1. 33/8 is just equal to 4 and 1/8.

33/8 is equal to 4 [tex]\frac{1}{8}[/tex] in terms of mixed fraction.

What are Fractions?

Fractions are numbers of the form [tex]\frac{p}{q}[/tex] where both p and q are real numbers. It can also be defined as a portion of a whole.

p is called the numerator and q is called the denominator.

There are different types of factors such as unit fractions, like fractions, unlike fractions, improper fractions, proper fractions, mixed fractions and so on.

Mixed fractions are fractions of the form q [tex]\frac{r}{d}[/tex] where q is the quotient, r is the remainder and d is the divisor upon dividing the numerator with the denominator of an improper fraction.

33/8 can be expressed in the mixed form by dividing 33 with 8.

We get quotient is 4 and remainder is 1.

So 33/8 = 4 [tex]\frac{1}{8}[/tex]

Hence 33/8 can be written as 4 [tex]\frac{1}{8}[/tex] in terms of mixed fraction.

To learn more on Fractions, click on the link given below:

https://brainly.com/question/10354322

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When using 3 letters to make an angle the middle letter is

Answers

X? Not sure what the answer choices are but try that

2. Construct the perpendicular bisector of CD. Label the intersection of CD and the perpendicular

bisector point M. (5 points)

From c to d line

I need help fast

Answers

Answer:

Step-by-step explanation:

A bisector is a line that divides an angle or a line into equal halves. In the case of a given line, the bisector will always be perpendicular to it.

To bisect CD, draw CD to the required length.

Open the compass to a radius greater than the the middle of CD.

With C as the center and the radius, describe an arc above and below CD. Repeat the same using D as the center.

Join the point of intersection of the two arcs by a line.

Label the point where the bisector passes through CD as M.

The construction would be as what is attached to this answer.

What does ¼ x (5.6) equal to

Answers

Answer:

The answer is 1.4 because you have to simplify it. Hope this helps :)

Step-by-step explanation:

Colin Davis machine company maintains a general ledger

Answers

Step-by-step explanation:

yf.bfz

vcg ,kgu nkkgfgvvxvh

y2-8+7
2x - y >4
Yes
No

Answers

No is the answer.
Thank you!


Find the area of the figure.
9 yd
6,4 yd

Answers

Answer:

144 should be your answer

Step-by-step explanation:

Write the expression: Five times the opposite of a number *

Answers

Answer:

5(-x)

Step-by-step explanation:

As a child gets older, will they be able to solve a puzzle faster? What is the dependent variable?

Answers

Answer:

the dependant variable is the time

BbhhhhhuskyNnnjhhhJndg.

Answers

.... What’s your question?

Need helppppppp!!!!Solve the absolute value equation: |3x - 6| = 12

Answers

Answer:

3x-6=12

3x-6=-12

x=6

x= -2

1st x = -2

2nd x = 6

Step-by-step explanation:

Find a perimeter of the bouncy castle find the area of the bouncy castle 2X squared minus 3X +7 times X squared +4

Answers

Answer:

The answer is below

Step-by-step explanation:

From the image attached, the length of the bouncy castle is X² +4 and the width is 2x² - 3x  +7

The bouncy castle is a rectangle. A rectangle is a quadrilateral in which opposite sides are equal and all the angles have equal measures. The diagonals of a rectangle also bisect each other. The perimeter of a rectangle = 2(length + breadth), Hence for the bouncy castle:

Perimeter of castle = 2(length + breadth) = 2(x² + 4 + (2x² - 3x  +7)) = 2(3x² - 3x + 11) = 6x² - 6x + 22

Perimeter of castle = 6x² - 6x + 22

The area of the castle = length × breadth = x² + 4 × (2x² - 3x  +7)) = 2x⁴ - 3x³ + 7x² + 8x² - 12x + 28 = 2x⁴ - 3x³ + 15x² - 12x + 28

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