Please help me. Use mathematical induction to prove the given statement for all positive integers n.
3+5+7+ ... + (2n+1)=n(n+2)
Part: 0/6
Part 1 of 6
Let P.be the statement: 3+5+7+ ... + (2n+1)=n(n+2).
Show that P, is true for n=
Answers
When proving by induction, your first step is to show that a given claim is true for some minimal case (often called the base case). The given statement is a claim about the sum of numbers of the form 2n + 1 where n is a positive natural number. The least positive natural number is n = 1, so that seems like a reasonable place to start.
Now, we show the claim holds for n = 1:
3 + 5 + 7 + … + (2n + 1) = n (n + 2)
is a statement about an arbitrary positive integer n; the case of n = 1 reduces to
3 = 1 (1 + 2) = 1 • 3 = 3
which is obviously true.
Next, we assume that the statement is true for some arbitrary positive integer n = k, so that
3 + 5 + 7 + … + (2k + 1) = k (k + 2)
This is called the induction hypothesis.
We then use this assumption to show that the claim is also true for the next positive integer, n = k + 1. This is the claim that
3 + 5 + 7 + … + (2k + 1) + (2 (k + 1) + 1) = (k + 1) (k + 1 + 2)
or
3 + 5 + 7 + … + (2k + 1) + (2k + 3) = (k + 1) (k + 3)
By the induction hypothesis, we have
3 + 5 + 7 + … + (2k + 1) + (2k + 3)
= k (k + 2) + (2k + 3)
= k² + 2k + 2k + 3
= k² + 4k + 3
= (k + 1) (k + 3)
and this is exactly what we wanted to show, that if P(k) is true, then P(k + 1) is also true.
This means we have
P(1) = true ⇒ P(2) = true ⇒ P(3) = true ⇒ …
and so on, and the claim is true for all positive integers n.
Related Questions
Find the prime factorization of 35. I need some explanation.
Answers
Answer:
factorize(35) = 5 x 7
Step-by-step explanation:
Solve for a, w, x, and y. x 31 w 60° 55° 46 a = W= x = y =
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Answer:
a 31. w 55. X 60 y 46
Step-by-step explanation:
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Hi help me please if you can
Answers
Answer:
Exact: 3 2/9
Estimated: 3
Step-by-step explanation:
1/3=3/9, so just convert it to that, then do some simple subtraction.
Show a Fraction name for 6
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Answer: 6/1
Step-by-step explanation:
please solve this....
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Answer:
I believe that's the solution!
sorry for my letters...
please answer number 41 and show work
Answers
Re-solve please.
2 5/6 + (-1 4/12) =
Answers
Answer:
1.5 as decimal, 1 1/2 as a fraction
Step-by-step explanation:
1. convert 2 5/6 & (-1 4/12) to a improper fraction.
17/6 and -16/12
2. multiply 17/6 by 2 to match -16/12's denominator
17/6 * 2 = 34/12
We now have
34/12 + -16/12
A positive number plus a negative number is the same thing as subtraction so we remove the negative sign and replace the plus with a minus sign.
We have:
[tex]\frac{34}{12}-\frac{16}{12}[/tex]
Now solve. 34-16 = 18
our answer is 18/12
Write an equation of the circle with center (-6, 2) and diameter 10.
Answers
Answer:
it something i know that much
Step-by-step explanation:
Solution:
We know that the formula to write the equation of a circle is (x – h)² + (y – k)² = r², where:
h = x coordinatek = y-coordinater = radius of circleCenter of circle: (-6,2)First, let's find the radius of the circle. We know that the diameter of the circle is ten units. The radius is half of the diameter. This means that the radius is five units because it is half the diameter, which is ten units.
Diameter: 10 unitsRadius: 10/2 units = 5 unitsCreating the equation:
(x – h)² + (y – k)² = r²=> (x + 6)² + (y – 2)² = 5²Hence, the equation to form a circle with center (-6, 2) and diameter 10 is (x + 6)² + (y – 2)² = 5².
Image attached~
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Suppose you are given a triangle with hypotenuse of length 2.5 and legs of length x−1 and x+1.
Answers
The value for the given right triangle of [tex]x[/tex] is [tex]\pm \sqrt{2}[/tex].
How to find the value of a variableBy Pythagorean theorem and some algebraic handling, we have the following second order polynomial:
[tex](x-1)^{2}+(x+1)^{2} = 2.5^{2}[/tex] (1)
Where [tex]x[/tex] is the desired variable.
[tex](x^{2}-2\cdot x +1)+(x^{2}+2\cdot x +1) = 2.5^{2}[/tex]
[tex]2\cdot x^{2}+2 = 2.5^{2}[/tex]
[tex]2\cdot x^{2}-4.25= 0[/tex]
[tex]x = \pm \sqrt{2}[/tex]
The value for the given right triangle of [tex]x[/tex] is [tex]\pm \sqrt{2}[/tex]. [tex]\blacksquare[/tex]
Remark
The statement is incomplete. Correct form is shown below:
Suppose you are given a triangle with hypotenuse of length 2.5 and legs of length [tex]x-1[/tex] and [tex]x+1[/tex]. Determine the value of [tex]x[/tex].
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Circle O and triangle OQR are shown below. What is the length of OQ?
48
64
O
48
80
R
112
64
Q
Answers
The required length of side OQ is given as 80 units, as of the given conditions.
What are Pythagorean triplets?In a right-angled triangle, its sides, such as hypotenuse, perpendicular, and base are Pythagorean triplets.
here,
From the figure,
Applying Pythagoras' theorem,
OQ² = OR² + RQ²
OQ = √[48² + 64²]
OQ = 80 units
Thus, the required length of side OQ is given as 80 units, as of the given conditions.
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Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Enter your answer in the box.
___°
Answers
Answer:
m<C = 62°
Step-by-step explanation:
[tex]m[tex]m [tex]x+20+3x=180[/tex]
[tex]4x=160,[/tex] [tex]x=40[/tex]
[tex]m [tex]118+m [tex]m [tex]m
How to solve this problem.
Answers
Answer:
ok so frog a travels at a rate of 1 ft every 30 seconds
which frog b travels 2 feet every second so
Frog A: 1/30
Frog B: 2/1 or 2
so Frog B will win
Hope This Helps!!!
Find the area of the shaded region.
Answers
Answer:
Area of the shaded region is 104.96 m².
Step-by-step explanation:
Firstly, finding the area of circle with radius 8 cm by substituting the values in the formula :
[tex]{:\implies{\tt{A_{(Circle)} = \pi {r}^{2}}}}[/tex]
[tex]{:\implies{\tt{A_{(Circle)} = 3.14{(8)}^{2}}}}[/tex]
[tex]{:\implies{\tt{A_{(Circle)} = 3.14{(8 \times 8)}}}}[/tex]
[tex]{:\implies{\tt{A_{(Circle)} = 3.14{(64)}}}}[/tex]
[tex]{:\implies{\tt{A_{(Circle)} = 3.14 \times 64}}}[/tex][tex]{:\implies{\tt{A_{(Circle)} = 200.96 \: {m}^{2} }}}[/tex]
Hence, the area of circle is 200.96 m².
[tex]\begin{gathered}\end{gathered}[/tex]
Secondly, finding the area of rectangle with length and breadth by substituting the values in the formula :
[tex]{:\implies{\tt{A_{(Rectangle)} = l \times b}}}[/tex]
[tex]{:\implies{\tt{A_{(Rectangle)} = 12 \times 8}}}[/tex]
[tex]{:\implies{\tt{A_{(Rectangle)} = 96 \: {m}^{2}}}}[/tex]
Hence, the area of rectangle is 96 m².
[tex]\begin{gathered}\end{gathered}[/tex]
Now, finding the area of shaded region by substituting the values in the formula :
[tex]{:\implies{\tt{A_{(Shaded)} = A_{(Circle)} - A_{(Rectangle)}}}}[/tex]
[tex]{:\implies{\tt{A_{(Shaded)} = 200.96 - 96}}}[/tex]
[tex]{:\implies{\tt{A_{(Shaded)} = 104.96 \: {m}^{2}}}}[/tex]
Hence, the area of shaded region is 104.96 m².
[tex]\rule{300}{2.5}[/tex]
Quadrilateral CAWT is the image of quadrilateral MOPS translated 7 units right and 5 units down. If you know the measure of ∠M, what other angle do you know the measure of?
∠C
∠A
∠W
∠T
Answers
i d k
Step-by-step explanation:
Option (A) will be the answer.
Rigid transformation: If the size and shape of a figure after transformation remains unchanged, transformation is called as rigid transformation.
(Example: Translation, Rotation and Reflection)
Given in the question,
Quadrilateral CAWT is the image of quadrilateral MOPS translated by 7 units right and 5 units down. Measure of ∠M is known.Since, image quadrilateral MOPS is formed by the translation of quadrilateral CAWT by 7 units right and 5 units down,
It's a rigid transformation having measures of all interior angles unchanged.
By the definition of rigid transformation → ∠M ≅ ∠C
Therefore, option (A) will be the answer.
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Find the equation of the line that contains the point (4,-2) and is perpendicular to the line y=-2x+8.
Answers
Answer:
[tex]\frac{1}{2} x - 4[/tex]
Step-by-step explanation:
Perpendicular lines will have slopes that are the multiplicative additive inverse of each other: a line with slope 3/4 is perpendicular to a line with slope -4/3, for example. You need a line that is perpendicular to a line with slope -2 (reading that from the -2 in the -2x portion of the given equation, which is written in slope-intercept form), so your new line must have slope +1/2.
With the slope and a point, we can come up with an equation using this formula:
y - y-coordinate = slope (x - x-coordinate)
So we have y - (-2) = 1/2 (x - 4).
Simplify the equation: y + 2 = 1/2 x - 2
Subtract 2 from both sides: y + 2 - 2 = 1/2 x - 2 - 2
Simplify: y = 1/2 x - 4.
a diver has to dive a depth of 200.2 ft go study
Answers
Answer:
The Diver has to dive 20.02 ft at every interval.
Step-by-step explanation:
10 intervals → 200.2
1 interval → 200.2/10
1 interval → 20.02
sanaya can do 1/3 part of a work in 1 day and chahanaa can do1/4 part of the same work in 1 day .How much work would they do working together
Answers
Answer: 36/5= 7.2 days to complete sanaya work day
Step-by-step explanation:
what is equavelent to 6(x+2y)+3+4y+5
Answers
Solve the equation
6x + 30 - 4 / 2
Answers
4/2 = 2
30-2= 28
6x + 28
Answer:
it's 6x+28
4 divide by 2 is 2
30 minus 2 is 28
Put together and hey 6x+28
Step-by-step explanation:
hope this helps
A normal distribution (X) has a mean of 100 and a standard deviation of 10.
What is the probability that X will be smaller than 80?
Use the Normal Table and keep all 4 decimal positions in your answer.
Answers
Using the normal distribution, it is found that there is a 0.0228 = 2.28% probability that X will be smaller than 80.
Normal Probability DistributionIn a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.In this problem:
The mean is of [tex]\mu = 100[/tex].The standard deviation is of [tex]\sigma = 10[/tex].The probability that X will be smaller than 80 is the p-value of Z when X = 80, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{80 - 100}{10}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a p-value of 0.0228.
0.0228 = 2.28% probability that X will be smaller than 80.
More can be learned about the normal distribution at https://brainly.com/question/24663213
An old road was 3.9 miles long. After a renovation it is 7.8 miles long. How many times as long is the road after the renovation than before?
Answers
Answer:
2 times. 3.9x2=7.8
Step-by-step explanation:
3+3=6
0.9+0.9=1.8
6+1.8=7.8
3.9x2=7.8
3+3=6
0.9+0.9=1.8
6+1.8=7.8
What is the value of the missing angle?
Answers
Answer:
x=60°
Step-by-step explanation:
The Sum of all sides of a pentagon is 540°
And in the figure there are 2 sides with 150° and 90°
Now we have,
[tex]90+90+150+150+x=540\\\\480+x=540\\\\x=540-480\\\\x=60[/tex]
Please help and tell me how you solved it
Answers
Answer:
63 degrees
Step-by-step explanation:
angle B and angle D are opposite interior angles, and therefore they have the same measurements. IM NOT SURE IF THIS IS THE CORRECT EXPLANATION, I LEARNED THIS A REALLY LONG TIME AGO AND SO I KIND OF FORGOT!!! SORRY
Will Give Brainliest
You Are Graphing Rectangle ABCD In The Coordinate Plane. The Following Are Three Of The Vertices Of The Rectangle A(3,5) B(3,−4) C(−2,−4)
Answers
Answer:
D(-2,5)
Step-by-step explanation:
Point D has the same x coordinate as point C and has the same y coordinate as the point A
Hi!
We want to find the x and y value for point D.
We can see that point C is the same amount to the left as point D. That means they will have the same x-value.
Furthermore, we can see that point A is the same amount up as point D. That means that they will have the same y-value.
We will use the given coordinates, (x, y).
-------------------------------------------------------------------------------------------------------------
The x-value of point C is -2, and the y-value of point A is 5.
That means that the coordinates of point D are (-2, 5).
Fund the volume of the cylinder. Round your answer to the nearest tenth.
Answers
Answer:
The volume of the cylinder is 785.4 cubic feet.
If 200,000 is saved at simple interest of 5%per annum,calculate the total amount after 4years
Answers
Answer:
$240000
Step-by-step explanation:
[tex]p = 200000[/tex]
[tex]t = 4[/tex]
[tex]r = 5 \: percent\: per \: anum[/tex]
[tex]si = \frac{p \times r \times t}{100} [/tex]
[tex] = \frac{200000 \times 5 \times 4}{100} [/tex]
[tex] = 2000 \times 5 \times 4[/tex]
[tex] = 2000 \times 20[/tex]
[tex] = 40000[/tex]
[tex]amount = si + principle[/tex]
[tex] = 200000 + 40000 = 240000[/tex]
HOPE IT HELPS YOU #ITZADMIRERThe total amount after 4 years is $240,000
What is the total amount?Simple interest rate is the interest that is paid only on the principal portion of a the amount that is saved.
Total amount = principal + interest
Interest = principal x time x interest rate
200,000 x 0.05 x 4 = $40,000
Total amount = 200,000 + 40,000 = $240,000
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Match the form with each linear Equation. Some matches will be used more than
once:
y - 6 = 10(2x - 1)
y = -9x + 7
y + 2 = 6(x-4)
y = 3x - 2
5x = 3y - 9
5x + 2y = 10 1) Standard Form. 2) Slope Intercept Form. 3) Point Slope Form. 4) None of the Forms.
Answers
Answer:
1) slope form
2) standard form
3)slope form
4)standard form
5) none of theese
6)point slope form
Step-by-step explanation:
It is possible to for a triangle with angles 25 degree 65 degree 90 degree
Answers
Answer:
yes. by angle sum property.
Step-by-step explanation:
did not understood the question well but hope its useful.
Evaluate 18 divided 2a if a = 3
Answers
Answer:
3
Step-by-step explanation:
[tex]\frac{18}{2a}[/tex]
a = 3
[tex]\frac{18}{2(3)}[/tex]
multiply 2 and 3
2 * 3 = 6
[tex]\frac{18}{6}[/tex]
simplify
18/6 = 3
Compare.
5 /3
8 /5
A. >
B.<
C. =
Answers
Answer:
a
Step-by-step explanation:
5/3 is 1.6666667
8/5 is 1.6
therefore 5/3 > 8/5