Enter the correct letter to match each summation expression with the property or formula.
n
Σ
i=1
cai is equal to
. a
n(n + 1)(2n + 1)
6
n
Σ
i=1
i is equal to
. b
n(n + 1)
2
n
Σ
i=1
c is equal to
. c c
n
Σ
i=1
ai
n
Σ
i=1
i3 is equal to
. d cn
n
Σ
i=1
i2 is equal to
. e
n(n + 1)
2
2
Answers
The correct letter that match each summation expression are;
[tex]\displaystyle \sum\limits_{i=1}^n[/tex] is equal to c
[tex]\displaystyle {\sum\limits_{i=1}^n i}[/tex] is equal to b
[tex]\displaystyle \sum_{i=1}^n c[/tex] is equal to d
[tex]\displaystyle\sum_{i=1}^n i^3[/tex] is equal to e
[tex]\displaystyle \sum\limits_{i=1}^n i^2[/tex]is equal to a
What is those the summation expression indicates?The summation expression indicates the sum of all values of a series of a function within a specified limit of values.
The equation for the sum of a constant c and the ith term of an of a sequence of numbers is equivalent to the product of the constant and the sum of the terms of the sequence as follows;
[tex]\sum\limits_{i=1} ^n {c\cdot a_i} = c\cdot \sum\limits_{i=1} ^n {a_i}[/tex] the correct option is c
The sum of ith terms from i = 1 to i = n is an arithmetic series with the formula;
Sₙ = (n/2)·[2·a + (n - 1)·d]
Where; a = The first term = 1
d = The common difference = 1
Therefore;
Sₙ = (n/2)·[2 + (n - 1)] = n + n·(n - 1)/2 = (2·n + n² - n)/2
(2·n + n² - n)/2 = (n² + n)/2 = n·(n + 1)/2
Therefore;
[tex]\sum\limits_{i=1}^n i = \dfrac{n\cdot (n + 1)}{2}[/tex] The correct option is b
The sum of n number of a constant, c is c·n, therefore;
[tex]\sum\limits_{i = 1}^n c = c\cdot n[/tex] The correct option is d
[tex]\sum\limits_{i=1}^n [(1 + i)^3-i^3[/tex] = (2³ - 1³) + (3³ - 2³) + ... + ((n + 1)³ - n³) = (n + 1)³ - 1³
(n + 1)³ - 1³ = n³ + 3·n² + 3·n
[tex]\sum\limits_{i=1}^n [(1 + i)^3-i^3[/tex] = n³ + 3·n² + 3·n
However, we get;'
[tex]\sum\limits_{i=1}^n [(1 + i)^3-i^3[/tex] = [tex]\sum\limits_{i=1}^n [ i^3 + 3\cdot i^2 + 3\cdot i+ 1-i^3][/tex]
[tex]\sum\limits_{i=1}^n [ i^3 + 3\cdot i^2 + 3\cdot i+ 1-i^3][/tex] = [tex]\sum\limits_{i=1}^n [ 3\cdot i^2 + 3\cdot i+ 1][/tex] = 3·S₂ + 3·S₁ + n
Where;
[tex]S_2 = \sum\limits_{i=1}^n i^2[/tex]
[tex]S_1 = \sum\limits_{i=1}^n i[/tex] = n·(n + 1)/2
[tex]\sum\limits_{i=1}^n [ 3\cdot i^2 + 3\cdot i+ 1][/tex] = n³ + 3·n² + 3·n = 3·S₂ + 3·S₁ + n
3·S₂ = n³ + 3·n² + 3·n - (3·S₁ + n) = n³ + 3·n² + 3·n - (3·(n·(n + 1)/2) + n)
n³ + 3·n² + 3·n - (3·(n·(n + 1)/2) + n) = n³ + 3·n²/2 + n/2
3·S₂ = n³ + 3·n²/2 + n/2
S₂ = n³/3 + 3·n²/6 + n/6 = (2·n³ + 3·n² + n)/6
S₂ = (2·n³ + 3·n² + n)/6 = n·(2·n² + 3·n + 1)/6
S₂ = n·(2·n² + 3·n + 1)/6 = n·(2·n + 1)·(n + 1)/6
S₂ = n·(2·n + 1)·(n + 1)/6
[tex]S_2 = \sum\limits_{i=1}^n i^2 = \dfrac{n\cdot (2\cdot n + 1)\cdot (n + 1)}{6}[/tex] The correct option is; a
(n + 1)⁴ - n⁴ = 4·n³ + 6·n² + 4·n + 1
Therefore;
(n + 1)⁴ - 1⁴ = 4·S₃ + 6·S₂ + 4·S₁ + n
4·S₃ = (n + 1)⁴ - 1⁴ - (6·S₂ + 4·S₁ + n)
(n + 1)⁴ - 1⁴- (6·(n·(2·n + 1)·(n + 2)/6) + 4·(n·(n + 1)/2) + n) = (n + 1)·(3·n² - n + 1)
4·S₃ = n²·(n + 1)²
S₃ = n²·(n + 1)²/4
[tex]S_3 = \sum\limits_{i=1}^n i^3 = \dfrac{n^2\cdot (n+1)^2}{4}[/tex] The correct option is e
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Related Questions
What is the measure of angle z in this picture? NEED HELP ASAP
Answers
137
Step-by-step explanation:
43+ z=180
z=180-43
= 137
Answer:
137°
Step-by-step explanation:
Subtract 43 from 180
180 is the degree of a straight line
Angles y and 43 degree angle are equal so to find z you subtract the 43 from 180
180 - 43 = 137
I need Help with this question. The screenshot is attached below. Help PLS
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Answer:
6
Step-by-step explanation:
True or False? When finding the average rate of change if an exponential function, the interval [3, 5] means we want to use the points when x = 3 and when y = 5.
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Answer:
yes
Step-by-step explanation:
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Explanation: multiply
Please answer correctly
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Explanation: Vertical Angles have the same measure
QRN= 65º
hope this helps!!
area of the painted part
Answers
Answer:
36
Step-by-step explanation:
7 + 7 + 11 + 11
Answer:
59.73 cm²
Step-by-step explanation:
[tex]2\pi r[/tex]
2 × π × 7
2 × 3.14 × 7
43.96 ÷ 4
10.99 cm²
11 cm - 7 cm = 4 cm
[tex]2\pi r[/tex]
2 × π × 4
2 × 3.14 × 4
25.12 ÷ 4
6.28 cm²
10.99 cm² + 6.28 cm² = 17.27 cm²
7 cm × 11 cm = 77 cm²
77 cm² - 17.27 cm²
59.73 cm²
The price for pumpkin last year is $1.40 per kilogram. The price for pumpkin this
year is increased by $0.80 per kilogram. What is the price for pumpkin this year?
Answers
Answer:
1.40×0.80=1.12 or 1.40/0.80=1.75
The number of bacteria is doubled every six hours. Estimate the number of bacteria present after 48 hours if initially there were 12 million bacteria present.
Answers
Answer:
96million
Step-by-step explanation:
48/6=8
12 000 000 * 8 = 96 000 000
=96 million
Answer:
96 million
Step-by-step explanation:
48/6= 8
8*12 million = 96 million
Can someone help pls
Answers
Answer:
F
Step-by-step explanation:
I'm guessing wjdhksdjene
Answer:
J
Step-by-step explanation:
The answer is option J
BRAINILIEST PLEASEwhat is the greatest common factor of 49x^2-21
Answers
Answer:
7x
Step-by-step explanation:
First find the common factors of the numbers: 49, 21.
That would be 1 and 7.
The GCF is 7 b/c it is greater than 1.
Find the common factors of x and x^2.
That is x times x
The factor for x ^1 is x itself.
The common factor for the both of them is x.
The GCF is then x.
The multiply the both of the GCFs (x.7) and you now have 7x.
So the GCF is 7x.
The greatest common factor of [tex]49\cdot x^{2}-21[/tex] is 7.
In this question, the greatest common factor of [tex]49\cdot x^{2}-21[/tex] is the greatest multiple applied on each term of the polynomial. First we notice that greatest common factor may be entirely constant. Then, we proceed to calculate each constant by factor decomposition:
[tex]49 = 7^{2}[/tex], [tex]21 = 3\times 7[/tex]
Hence, we conclude that greatest common factor of [tex]49\cdot x^{2}-21[/tex] is 7.
We kindly invite you to check question on greatest common factor: https://brainly.com/question/11221202
Someone help please!!! I don't understand. (20 points)
Answers
Fill in the table using this function rule.
y = 2x + 5
Answers
First one x = 1
y = 2(1) + 5
y= 7
I’ll tell u all the answers here in order:
7
9
15
19
Answer:
The answer is
[ X ][ Y ]
[ 1 ][ 7 ]
[ 2 ][ 9 ]
[ 5 ][ 15 ]
[ 7 ][ 19 ]
Brainiest please and lot of likes:)
find the area of the figure. use 3.14 for pi
Answers
Answer:
33.29
Step-by-step explanation:
A=πr^2
A=π2^2=12.57
12.57/2=6.29
7-4=3
3x2=6
6/2=3
4x6=24
6.29+3+24=33.29
Semi-Circle:
Area = 0.5*(pi*radius^2) = 0.5(3.14*2^2) = 0.5(3.14*4)
= 6.28 m^2
Rectangle:
Area = length*width = 6*4 = 24 m^2
Triangle:
Area = 0.5(Base*Height) = 0.5(3*2) = 3 m^2
Total Area = 6.28 + 24 + 3 = 33.28 m^2
Which of the following is an arithmetic sequence?
© 0, 2, 4, 6, ...
O 1, 2, 4, 8, ...
O 1, 4, 8, 13, ...
Answers
Arithmetic sequence formula is a1 + (n - 1)d
Where a1 is the first term in the sequence, and d is the difference between the two terms.
Find the third term in the first sequence with the equation: 0 + (3 - 1)(4 - 2) = 2(2) = 4 ✅
Fourth term: 0 + (4 - 1)(6 - 4) = 3(2) = 6 ✅
The others don’t work because their differences are not consistent
1.
35, 56, 34, 44, 52, 12, 34, 45
Mean:
Median:
Mode:
Answers
Answer:
Mean x¯¯¯ 39
Median x˜ 39.5
Mode 34
Step-by-step explanation:
Mean: 39
Median: 39.5
Mode: 34
What is the image of (-4,-4) after a dilation by a scale factor of 3 centered at the
origin?
Answers
Answer:
(-12, -12)
Step-by-step explanation:
Since the scale factor is 3, you would multiply the x-coordinate and y-coordinate by it.
-4×3=-12
Hope this helps!!
I need help I just got this today and I don’t know what to do btw it’s due tomorrow
Answers
Answer:
x = 35
Step-by-step explanation:
The proper equation of proportions is
x 42
----- = ------ and it yields x = 35
40 48
Answer:
? = 35
Step-by-step explanation:
The lengths of corresponding sides of similar polygons are in the same ratio.
The sides measuring 48 and 40 are corresponding sides.
The ratio of their lengths is 48/40.
The sides measuring 42 and ? are also corresponding, so they are in the same ratio.
That means we can set up a proportion to solve for the side with length ?.
48/40 = 42/?
Simplify the fraction on the left side.
6/5 = 42/?
Cross multiply.
6 * ? = 5 * 42
Divide both sides by 6.
? = 5 * 7
? = 35
10x-18=15x+48 solve for the x please.
Answers
Answer: 15
Step-by-step explanation:
What is true about the completely simplified sum of the polynomials 3x^2y^2-2xy^5 and -3x^2y^2 + 3x^2y^2 + 3x^4y?
Answers
Given:
The polynomials are:
[tex]3x^2y^2-2xy^5[/tex]
[tex]-3x^2y^2+3x^2y^2+3x^4y[/tex]
To find:
The completely simplified sum of the polynomials.
Solution:
We have,
[tex]3x^2y^2-2xy^5[/tex]
[tex]-3x^2y^2+3x^2y^2+3x^4y[/tex]
The sum of given polynomials is:
[tex]Sum=3x^2y^2-2xy^5-3x^2y^2+3x^2y^2+3x^4y[/tex]
[tex]Sum=-2xy^5+3x^4y+3x^2y^2[/tex]
Therefore, the sum of the given polynomials is [tex]-2xy^5+3x^4y+3x^2y^2[/tex]. It is a polynomial with degree 6 and leading coefficient -2.
Find the perimeter of the pentagon MNPQR with vertices M(2, 4), N(5, 8), P(8, 4), Q(8, 1), and R(2, 1)
Answers
Answer:
The pentagon MNPQR has a perimeter of 22 units.
Step-by-step explanation:
Geometrically speaking, the perimeter of the pentagon is the sum of the lengths of each side, that is:
[tex]p = MN + NP + PQ + QR + RM[/tex] (1)
[tex]p = \sqrt{\overrightarrow{MN}\,\bullet \, \overrightarrow{MN}} + \sqrt{\overrightarrow{NP}\,\bullet \, \overrightarrow{NP}} + \sqrt{\overrightarrow{PQ}\,\bullet \, \overrightarrow{PQ}} + \sqrt{\overrightarrow{QR}\,\bullet \, \overrightarrow{QR}} + \sqrt{\overrightarrow{RM}\,\bullet \, \overrightarrow{RM}}[/tex] (1b)
If we know that [tex]M(x,y) = (2,4)[/tex], [tex]N(x,y) = (5,8)[/tex], [tex]P(x,y) = (8,4)[/tex], [tex]Q(x,y) = (8,1)[/tex] and [tex]R(x,y) = (2,1)[/tex], then the perimeter of the pentagon MNPQR is:
[tex]p =\sqrt{(5-2)^{2}+(8-4)^{2}} + \sqrt{(8-5)^{2}+(4-8)^{2}}+\sqrt{(8-8)^{2}+(1-4)^{2}}+\sqrt{(2-8)^{2}+(1-1)^{2}}+\sqrt{(2-2)^{2}+(4-1)^{2}}[/tex][tex]p = \sqrt{3^{2}+4^{2}} + \sqrt{3^{2}+(-4)^{2}}+\sqrt{0^{2}+(-3)^{2}}+\sqrt{(-6)^{2}+0^{2}}+\sqrt{0^{2}+3^{2}}[/tex]
[tex]p = 22[/tex]
The pentagon MNPQR has a perimeter of 22 units.
The pentagon MNPQR has a perimeter of 22 units.
Geometrically speaking, the perimeter of the pentagon is the sum of the lengths of each side, that is:
p = MN + NP + PQ + QR + RMp=MN+NP+PQ+QR+RM (1)
p = \sqrt{\overrightarrow{MN}\,\bullet \, \overrightarrow{MN}} + \sqrt{\overrightarrow{NP}\,\bullet \, \overrightarrow{NP}} + \sqrt{\overrightarrow{PQ}\,\bullet \, \overrightarrow{PQ}} + \sqrt{\overrightarrow{QR}\,\bullet \, \overrightarrow{QR}} + \sqrt{\overrightarrow{RM}\,\bullet \, \overrightarrow{RM}}p=
MN
∙
MN
+
NP
∙
NP
+
PQ
∙
PQ
+
QR
∙
QR
+
RM
∙
RM
(1b)
If we know that M(x,y) = (2,4)M(x,y)=(2,4) , N(x,y) = (5,8)N(x,y)=(5,8) , P(x,y) = (8,4)P(x,y)=(8,4) , Q(x,y) = (8,1)Q(x,y)=(8,1) and R(x,y) = (2,1)R(x,y)=(2,1) , then the perimeter of the pentagon MNPQR is:
p =\sqrt{(5-2)^{2}+(8-4)^{2}} + \sqrt{(8-5)^{2}+(4-8)^{2}}+\sqrt{(8-8)^{2}+(1-4)^{2}}+\sqrt{(2-8)^{2}+(1-1)^{2}}+\sqrt{(2-2)^{2}+(4-1)^{2}}p=
(5−2)
2
+(8−4)
2
+
(8−5)
2
+(4−8)
2
+
(8−8)
2
+(1−4)
2
+
(2−8)
2
+(1−1)
2
+
(2−2)
2
+(4−1)
2
p = \sqrt{3^{2}+4^{2}} + \sqrt{3^{2}+(-4)^{2}}+\sqrt{0^{2}+(-3)^{2}}+\sqrt{(-6)^{2}+0^{2}}+\sqrt{0^{2}+3^{2}}p=
3
2
+4
2
+
3
2
+(−4)
2
+
0
2
+(−3)
2
+
(−6)
2
+0
2
+
0
2
+3
2
p = 22p=22
The pentagon MNPQR has a perimeter of 22 units.
Use the graph to estimate the selling price for a 4-year old car.
O A. $24,000
B. $20,000
che c. $8,000
D. $16,000
Answers
Answer:
The Correct answer would be option D, $16,000.
Step-by-step explanation:
Look at the bottom section of the graph, that is labled
"Age (in years)" at the number 4 line (representing the 4 year old car), and find the nearest dot on that line of the graph, it goes directly to 16,000.
Find the volume of the figure
HELP ME PLEASE IM STRUGGLING!!
Answers
Answer:
I got great news for you! I have a wonderful grade in math...
Anyway the answer is 6,318
Step-by-step explanation:
Ok so 18x13 is 234. 234x21 is 4,914. Now we have the volume of the box (4,914).
as we know 18x13 is 234. So if we multiply 234x12 we get 2,808. So now all we have to do is divide 2,808 by 2. That gives us 1,404.
1,404+4,914=6,318
So now WALLA we have our answer.
Determine o valor da expressão: -1 + 6 x (7 – 4 ÷ 2)
Answers
Answer:
29 is the correct answer. Hope you like my answer.
What is the multiplication sentence for the following diagram?
Answers
Answer:
welllllllll thats is the confusion diagram
Boubacar went to the store to buy some walnuts. The price per pound of the walnuts
is $6.50 per pound and he has a coupon for $3.75 off the final amount. With the
coupon, how much would Boubacar have to pay to buy 4 pounds of walnuts? Also,
write an expression for the cost to buy p pounds of walnuts, assuming at least one pound is purchased.
Answers
Answer:
Step-by-step explanation:
Price per pound of the walnuts = $ 6.50
Price of 4 pounds of the walnuts = 4 *6.50 = $ 26
Amount needs to be paid = Price of 4 pounds of walnut - coupon amount
= 26 - 3.75
= $ 22.25
Expression:
Cost to buy p pounds of walnuts = p * 6.50 = 6.5p
Cost to buy p pounds of walnuts, assuming at least one pound is purchased = 6.50 + 6.5p
The talk-time battery life of a group of cell phone is normally distributed with a mean of 5 hours and a standard deviation of 15 minutes. A) What percent of the phones have a battery life of at least 4 hours and 45 minutes?
Answers
The correct answer is 84%
The phone battery lasts at least 4 hours and 45 minutes is 84%.
What is normal distribution of data?A normal distribution is a probability distribution that is perfectly symmetrical around its mean (center), showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.
For the given situation,
The normally distribution has a mean of 5 hours and a standard deviation of 15 minutes.
50% of the normal distribution lies to the right of the mean. So, the battery will last longer than 5 hours.
The interval from 4 hours and 45 minutes and 5 hours represents one standard deviation to the left of the mean.
So, about 34% of time, the battery will last between 4 hours and 45 minutes and 5 hours.
Thus, the probability that the battery lasts at least 4 hours and 45 minutes,
⇒ [tex]50+34[/tex]
⇒ [tex]84[/tex]
Hence we can conclude that the phone battery lasts at least 4 hours and 45 minutes is 84%.
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Choose the correct future tense form of the verb in parentheses.
Ellos (saber) la verdad.
sabré
sabrás
sabremos
sabrán
.
Answers
Answer:
The correct future tense would be sabrán, which basically means they will know in English.
Step-by-step explanation:
Since the question states "Ellos" (they) as in a group of people, sabrán would be the right one.
These words (in order) basically roughly mean
I will know
You will know
We will know
THEY will know
Answer:
D. sabrán
Step-by-step explanation:
The form of the verb, saber, has to match the form of the pronoun that is doing the action. "Ellos" is doing the action, and ellos means "they". The "they" form of a verb ends with an "en" or "an". So your best answer is sabrán.Please click the heart if this helped.
Sahiti has a circular tabletop with a 2-foot diameter. She wants to paint a pattern on the table top that includes a 2-foot-by-1-foot rectangle and 4 squares with sides 0.5 foot long. Which information makes this scenario impossible?
A. There will be no room left on the tabletop after the rectangle has been painted.
B. A 2-foot-long rectangle will not fit on the circular tabletop.
C. Squares cannot be painted on the circle.
D. There will not be enough room on the table to fit all the 0.5-foot squares.
Answers
If a single fair die is rolled, find the probability of rolling a 3 given that the number is prime.
Answers
Answer:
1/3 chance
Step-by-step explanation:
2,3, and 5 are prime
1 is NOT a prime number
-4(m + 18) = 8 answer
Answers
Answer:
m = -20
Step-by-step explanation:
first distribute:
-4(m + 18) = 8
-4m - 72 = 8
Now add 72 to both sides to get m by itself
-4m = 80
and divide by -4 on both sides
m = -20
Hope this helped! :)
Answer: m = -20
Step-by-step explanation:
−4(m + 18) = 8
Step 1: Simplify both sides of the equation.
−4(m + 18) = 8
(−4)(m) + (−4)(18) = 8 (Distribute)
−4m + −72 = 8
−4m − 72 = 8
Step 2: Add 72 to both sides.
−4m − 72 + 72 = 8 + 72
−4m = 80
Step 3: Divide both sides by -4.
−4m / − 4 = 80 / −4
m = −20