Answers
Recall the binomial theorem:
[tex]\displaystyle (a+b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k[/tex]
a. Let a = b = 1 and n = 8. Then the sum above gives the sum shown here,
[tex]\displaystyle \sum_{k=0}^8 \binom8k 1^{n-k} 1^k = \sum_{k=0}^8 \binom8k[/tex]
so it reduces to (1 + 1)⁸ = 2⁸ = 256.
b. Let a = 1 and b = -1 and k = 8. Then by the same argument, the sum reduces to (1 - 1)⁸ = 0.
c. Notice that
[tex]\displaystyle \sum_{j=0}^n \binom n{n-j} = \binom nn + \binom n{n-1} + \binom n{n-2} + \cdots \binom n2 + \binom n1 + \binom n0[/tex]
but this is nearly identical to the sum in part a, with a = b = 1 and arbitrary n, but the order of terms is reversed. Then it reduces to (1 + 1)ⁿ = 2ⁿ.
d. Let a = 1 and b = 2. Then the sum is (1 + 2)ⁿ = 3ⁿ.
Related Questions
Can someone please help me on this one ?
You are considering different investment strategies to save for your retirement.
Option 1: You invest $50/ month at a rate of 3.5% APR compounded monthly for 35 years.
Option 2: You invest $150/ quarter at a rate of 4.15% APR compounded monthly for 35 years.
Option 3: You invest $4,000 at a rate of 5.5% APR compounded monthly for 35 years.
Complete the table below and answer the questions below it.
You may use this calculator http://www.mycalculators.com/ca/savecalcm.html to help you.
(Hint: The “Present Value” for options 1 and 2 is 0 while for option 3 it is $4,000.00. Also, the
top radio button in the Contributions box should be selected so that contributions are made at the beginning of the period.)
Table:
Option # | Contributions | Final Balance | Total Interest Earned
1
2
3
1) How did you calculate the contribution for each option? In which option would you
contribute the least?
2) Which option yielded the highest amount at the end of the 35 years and what was the basis of the plan?
3) What is the difference in the principal invested for the highest and lowest final
balances? What is the difference in the interest earned?
4) What plan would you recommend for the largest return on investment? (This means
the most interest for the least amount invested)
5) How does the differing interest rates in each option play into your recommendation?
I would appreciate full feedback! Thank you :)
Answers
Return on investment increases as the interest rate applied, the duration of investment, and the principal invested increases.
The responses are as follows;
1) Total contribution = Single contribution Value × Number of contributions2) Option 23) $-4,0004) Option 35) The higher interest rate increases the return on investmentReasons:
Option 1: Investment
The present value = 0
The contributions made = $50 monthly
APR of investment = 3.5%
Number of years of investment = 35 years
The contribution = $50 × 12 × 35 = $21,000
Future value (final balance) = $41,230.22
Interest Earned = $41,230.22 - $21,000 = $20,230.22
Option 2: Investment
The present value = 0
The contributions made = $150 per quarter
APR on investment = 4.15%
Number of years of investment = 35 years
The contribution = $150 × 4 × 35 = $21,000
Future value = $47,503.88
Interest Earned = $47,503.88 - $21,100
Option 3: Investment
Present value = $4,000
APR on investment = 5.5%
Number of years of investment = 35 years
The contribution = $4,000
Future value = $27,300.26
Interest Earned = $27,300.26 - $4,000 = $23,300.26
[tex]\begin{array}{|c|c|c|c|}Option \ number&Contributions&Total \ Interest \ Earned&Final \ Balance\\1&\$50&\$20,230.22&\$41,230.22\\2&\$150&\$26,403.88&\$47,503.88\\3& \$4,000 &\$23,300.26 &\$27,300.26 \end{array}\right][/tex]
1) The total contribution is given by the product of the monthly contribution and the number of timer units.
2) The option that yielded the highest amount at the end of 35 years is the option 2.
The basis of the plan is quarterly investment of $150 and an annual interest rate of 4.15% compounded monthly.
3) The principal with the highest final balance is the principal for Option 2. which is 0
The principal that has the lowest final balance is the principal for option 3 which is $4,000
The difference in the principal for the highest and lowest final balances is
$0 - $4,000 = $-4,000
4) The plan recommended for the largest return on investment is option 2, in which the amount invested increased by more than 600%
5) The higher the interest rate, the higher the return on the amount invested.
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A microscope shows you an image of an object that is 80 times the object's actual size. Therefore, the scale factor of the enlargement is 80. An insect has a body length of 7 millimeters. What is the body length of the insect under the microscope?
A. 56 millimeters
B. 560 centimeters
C. 5,600 millimeters
D. 560 millimeters
Answers
Answer:
560 mm (Answer D)
Step-by-step explanation:
Multiply the 7 mm measurement by 80; the result is 560 mm (Answer D).
WHAT IS 159 DIVIDED BY 3 (ITS NOT HARD PLEASE ANSWER I DON'T HAVE A CACULATER)
Answers
Answer:
Step-by-step explanation:
The asked quotient of 159 divided by 3 is the number 51.
What is divide?It is the inverse of multiplication, and it is used to find out how many times one number can be divided by another. In mathematics, the division is an arithmetic operation that involves splitting a quantity into equal parts or groups.
Here,
As mentioned in the question, we have to determine the quotient of 159 divided by 3
So let the quotient be y,
y = 159/3
y = 51
Thus, the asked quotient of 159 divided by 3 is number 51.
Learn more about divide here:
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If you randomly select a card from a well-shuffled standard deck of 52 cards, determine the probability that the card you select is not a spade.
A. Write your answer as a simplified fraction.
B. Write your answer as a decimal, rounded to the nearest thousandth.
C. Write your answer as a percent. Round to the nearest tenth of a percent and don't forget to include a percent sign, %, in your answer.
Answers
Answer:
a. 1/13
b. 0.077
c. 7.7
Step-by-step explanation:
a. there are 4 of the 6 in a deck of cards so the probability of taking this cards from the deck of 52 cards is 4/52. Given you 1/13.
Help me please ASAP
!!!!!
Answers
Answer:pretty sure the -4 would be on the right of the seven. And to the left of -1.
Step-by-step explanation:
THIS QUESTION IS WORTH 20 PTS!!! WILL REWARD BRAINLIEST!!!
Convert the exponential equations into logarithmic form.
Answers
Answer:
[tex] \: [/tex]
1. C = 81 , D = 4
2. E = 625, F = 4
3. G = 10,000 , H = 4
4. J = 0.01 , K = -2
Hope it helps you ™
A scientist needs 180 milliliters of a 20% acid solution for an experiment. The lab has available a 25% and a 10% solution. How many milliliters of the 25% solution and how many milliliters of the 10% solution should the scientist mix to make the 20% solution
Answers
9514 1404 393
Answer:
120 mL of 25% solution60 mL of 10% solutionStep-by-step explanation:
Let q represent the quantity of 25% solution in the mix. Then 180-q is the quantity of 10% solution. The quantity of acid in the mix is ...
0.10(180 -q) +0.25(q) = 0.20(180)
18 +0.15q = 36 . . . . simplify
0.15q = 18 . . . . . . . . subtract 18
q = 18/0.15 = 120 . . . . divide by the coefficient of q
180-q = 60
The scientist should mix 120 mL of 25% solution with 60 mL of 10% solution.
1. Which of the following is an example of acceleration? A. 10m/s B. 64km/h/min C. 50 cm2/s2 D. 46 km/h
Answers
Answer:
i beleive c for the proper answer form is m/s 2 c is the only one to show to square anything. if that's wrong the a but there is no square symbol.
Find the value of y for the given value of x.
y = x/2 + 9;x = -12
Answers
Step-by-step explanation:
x = -12
y = x/2 + 9
y = -12/2 + 9
y = -6 + 9
y = 3
Which function has an inverse this is a function?
b(x)=x^2 + 2x
d(x)=-4x
m(x)=-3
p(x)=|x|
Answers
Answer :-
Option 2
Step by step explanation :-
d (x) = -4x has an inverse .
Given: D is the midpoint of AC, ED = FD, and angle EDA = angle FDC
Prove that: Triangle AED = Triangle CFD
Answers
Answer:
Step-by-step explanation:
Side-Angle-Side postulate (SAS)
There are ΔAED and ΔCFD triangles that are congruent due to the SAS congruence postulate.
What are congruent triangles?Two triangles are said to be congruent if their corresponding sides and angles are equal.
As per the given question,
D is the midpoint of AC, ED = FD, and angle EDA = angle FDC
In ΔAED and ΔCFD,
Side DE = Side DF
m∠EDA = m∠FDC
Side AE= Side FC
According to SAS, two triangles are congruent if the two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle.
Thus, both triangles are congruent.
Hence, there are ΔAED and ΔCFD triangles that are congruent due to the SAS congruence postulate.
Learn more about congruent triangles here:
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A sample in which each element in a population has a equal chance of being selected
Answers
Answer:
simple random sample
Step-by-step explanation:
I hope that's it
A ruler 30 cm long is broken into two parts in the ratio 8:7. How long
are the two parts?
Answers
Answer:
Step-by-step explanation:
Length of each part of the ruler will be
8x and 7x
. Hence, 8x+7x=30 => 15x=30 =>x=2
Hence, the lengths will be
8x=8×2=16
7x=7×2=14
(BEST AND FASTEST GETs BRAINLIEST)Michael works at a furniture store. He makes commission for every piece of furtniture he sells. He sold a couch for $1245 he got a check for $62.25. What was the % of commission he made?
Answers
Step-by-step explanation:
if 1425 then the answer is 100
if it's 62.25 then 77 percent
what is 250% of 5
please help
Answers
Answer:
12.5
Step-by-step explanation:
You can read this problem as ...
2.5 × 5 = 12.5
__
You can think of % as meaning /100. You can think of "of" as meaning "times". Then the expression
250% of 5
can be thought of as ...
250/100 × 5
= 1250/100 = 12.5
1. You are interested in recycling behavior and whether it varies between urban and
rural residents. You know that the statewide recycling rate is 48.89%, and you
survey 140 rural residents and 310 urban residents to find out if they recycle.
180 urban residents respond that yes, they recycle. 130 do not. 40
rural residents recycle, and 100 do not.
a) Construct an expected frequencies crosstab, based on the sample size and the
statewide recycling rate.
b) Construct an observed frequencies crosstab, based on your actual data.
(NOTE for 1a and 16: Be sure to include both percentages and frequency values, in
Answers
Answer:
65
Step-by-step explanation:
Given the inequality y<=12x+4, which of the following statements are
correct? Check all that apply.
A. The line should be dashed.
B. Shading should go above the line.
C. The line should be solid.
D. Shading should go below the line.
Answers
Answer:
c)the line should be solid
As per <= sign the line should be solid
Next
Let's isolate x and y
[tex]\\ \sf\longmapsto y\leqslant 12x+4[/tex]
[tex]\\ \sf\longmapsto 12x-y\geqslant 4[/tex]
Put (0,0)[tex]\\ \sf\longmapsto 0-0\geqslant 4[/tex]
Yes it's trueShading should go below the line towards origin
Up arrow, right arrow, down arrow, left arrow, up arrow, right arrow, down arrow, left arrow. What's the conjecture
Answers
Answer:
The correct option is;
If one reflects a figure across the y-axis, the points of the image can be found using the pattern (x, y) Right-arrow (x, -y).
If one reflects a figure across the y-axis, the points of the image can be found using the pattern (x, y) Right-Arrow (-x, y).
Taking the result from the first reflection (x, -y) and applying the second mapping rule will result in (-x, -y), not (y, x), which reflection across the line y = x should give
Step-by-step explanation:
We have that for reflection across the x-axis, (x, y) → (x, -y)
For reflection across the y-axis, (x, y) → (-x, y)
Therefore, given that the pre-image before the reflection across the y-axis is (x, -y), we have;
For reflection across the y-axis, (x, -y) → (-x, -y)
For reflection across the line, y = x, gives (x, y) → (y, x) which is not the same as (-x, -y)
A car travels at an average speed of 56 miles per hour how many miles does it travel in 5hrs and 30min
Answers
Answer:
308 miles
Step-by-step explanation:
speed * time = distance
56 * 5.5 = 308
308 miles
WILL GIVE BRAINLIEST
These two triangles are congruent by ....
O SSS
O SAS
O HL
O ASA
O AAS
O the triangles are NOT congruent
Answers
SAS postulate states that if two sides and one angle of the two triangles are congruent then the corresponding triangles are congruent.
These two triangles are congruent by SAS postulate
Proof
Statements Reasons
In ΔNHJ ↔ ΔKHJ
mHJ≅mHJ common to both Δs∠N≅∠K right anglesmHN≅mHK givenFrom 1,2, and 3 ΔNHJ ≅ ΔKHJ SAS≅SAS
For more explanation click
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https://brainly.com/question/6422888
Round to the nearest cent 0.2745
Answers
Ten-thousandths: 0.2745
Thousandths: 0.275
hundredths: 0.27
Tenths: 0.3
Part of a city map is shown. If m 5 = x, which angles also have a measure of x? Select all that apply.
A 1
B 4
C 9
D 12
Thanks!!!
Answers
Answer:
C 9
D 12
Step-by-step explanation:
9 is a corresponding angle to 5 and 12 is an opposite angle to 9
Also by convention, grid lines are at right angles to each other so 5, 6, 7, 8, 9, 10, 11, and 12 are all equal at x = 90°
help me someone please
Answers
Answer:4root3
Step-by-step explanation:
if you dont know what a equilateral triangle its just. a triangle basically
(im in 7th btw)
I need help with part a and b please
Answers
Answer:
The total answer should be 38 on the top
Step-by-step explanation:
a) So the first partial quotient would be equal to 30 as 31 x 30 = 930
b) Second partial quotient would be 8 as 31 x 8 = 248
Answer:
0
38
I hope it's help byeeee
2y⁴×5y³
Send a message
Answers
Answer:
10y^7
Step-by-step explanation:
Answer:4+3=7
5×2=10
Step-by-step explanation:
D. Directions: Write the equation for line in any form) for the following points
(0,4) and (-3,12)
Answers
Answer:
y = -8/3x + 4
Step-by-step explanation:
The simple interest charged on a 12-month loan of $82,000 is $631. Find the simple interest rate. (Round your answer to one decimal place.)
Answers
Step-by-step explanation:
631/82000×100
=0.8
MARK ME BRAINLIEST PLSS!
What is the slope-intercept form of an equation of a line that passes through the point (0,-1) and is perpendicular to 5x-y=3?
Answers
Answer:
y = -1/5x - 1
Step-by-step explanation:
Perpendicular lines slopes are negative reciprocals of each other.
Write the given equation in the slope-intercept form: y = mx + b
5x - y = 3
-y = -5x + 3
y = 5x - 3 multiply by -1
The slope of the given line is 5 ( the coefficient of the x-term).
The slope of the new equation will -1/5 (this is the negative reciprocal of 5).
Next find "b" using y = mx + b, the new slope -1/5 and the point (0, -1)
y = mx + b
-1 = -1/5(0) + b
-1 = 0 + b
-1 = b
Now write the equation y = -1/5x - 1
Aaron creates a stacked cake comprised of three cylindrical sections as pictured below, and plans to cover the top of each section with a different color icing. Each lower section is a total of two inches wider than the section above it. If the yellow top of the upper section has an area of 12.56 square inches, what is the approximate area of the green visible portion of the bottom section which he will need to cover with icing?
Answers
Answer:
Step-by-step explanation:
A = πR²
top layer has a radius of
R = √(12.56/π) = 2 inches
The middle layer has a radius of 3 inches and an area of
A₂ = π3² = 9π in²
The bottom layer has a radius of 4 inches and an area of
A₃ = π4² = 16π in²
so the visible area on the bottom layer has an area of
16π - 9π = 7π in² or about 22 in²
21.2x+c is equivalent to 5.3(4x-2.6)
Answers
Yooooo what up how u doin ?
[tex]21.2x + c = 5.3(4x - 2.6)[/tex]
[tex]21.2x + c = 21.2 x - 13.78[/tex]
Subtract both sides 21.2x
[tex]21.2x - 21.2x + c = 21.2x - 21.2x - 13.78 \\ [/tex]
[tex]c = - 13.78[/tex]
There u go.... Don't have a good day , Have a great day ❤
