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
Contact [7]
3 years ago
14

Show that n^2-1 is divisible by 8,if 'n' is an odd positive integer.

Mathematics
1 answer:
Marina86 [1]3 years ago
6 0
n-an\ odd\ positive\ integer\ \ \ \Rightarrow\ \ \ n=2k+1\ \ \ and\ \ \ k\in Natural\\\\n^2-1=(2k+1)^2-1=(2k)^2+2\cdot2k\cdot1+1^2-1=4k^2+4k=\\\\=4k(k+1)\\\\if\ k-odd\ \ then\ (k+1)-even,\ \ \ then\ k(k+1)-even\\\\\ \ \ \Rightarrow\ \ \ k(k+1)=2m\ \ \ and\ \ \ m\in N\ \ \ \Rightarrow\ \ \ 4k(k+1)=4\cdot2m=8m\\\\\\if \ k-even,\ \ \ then\ k(k+1)-even\\\\\ \ \Rightarrow\ \ \ k(k+1)=2p\ \ \ and\ \ \ p\in N\ \ \ \Rightarrow\ \ \ 4k(k+1)=4\cdot2p=8p\\\\

------------------------\\\\ for\ each\ k \in N\ the\ product\ of\ \ \  k (k +1)\ \ \  is\ even,\ so:\\\\if\ 'n'\ is\ an \ odd\ positive\ integer,\ then\ \ \ n^2-1\ \ \ is\ divisible\  8.
You might be interested in
Janey bought a used bicycle for $640. the bike was 3 years old when she bought it, and cost $850 new. assuming it decreases in v
Radda [10]
The first thing we must do in this case is to define the variables.
 We have then:
 v: the value of the bicycle
 t: years
 The standard equation of the line is:
 v-vo = m (t-to)
 The slope is:
 m = (v2-v1) / (t2-t1)
 Substituting values:
 m = (640-850) / (3-0)
 m = -70
 We choose an ordered pair:
 (to, vo) = (0, 850)
 Substituting values:
 v-850 = -70 (t-0)
 Rewriting:
 v = -70t + 850
 Answer:
 
assuming it decreases in value by the same amount each year, a linear equation for the value of the bicycle, v, when it is t years old is:
 
v = -70t + 850
6 0
3 years ago
a triangular prism has an isosceles right triangular base with a hypotenuse of square root of 32 and a prism height of 12. a squ
omeli [17]

Answer:

The lengths of the sides of the square base are  2\sqrt{2}\ units

Step-by-step explanation:

<u><em>The correct question is</em></u>

A triangular prism has an isosceles right triangular base with a hypotenuse of square root of 32 and a prism height of 12. a square prism has a height of 12 and its volume is equal to that of the <em><u>triangular</u></em> prism. what are the dimensions of the square base, in simplest radical form?

we know that

The volume of a prism is given by the formula

V=Bh

where

B is the area of the base

h is the height of the prism

so

If the volumes of the solids are the same and their  heights are the same, then the area of the base of the triangular prism must be equal to the area of the base of the square prism

step 1

Find the area of the base of the triangular prism

Remember that in a right isosceles triangle, the two legs are equal

Let

x ---> the length side of each leg

Applying the Pythagorean Theorem

(\sqrt{32})^2 =2x^2

x^2=32/2\\x^2=16\\x=4\ units

The area of the triangular base is equal to

A=\frac{1}{2}(4)(4)=8\ units^2

step 2

Find the dimensions of base of the square prism

Let

b ---> the length side of the square base

The area of a square is given by the formula

A=b^2

we have

A=8\ unit^2 ----> is the same that the area of triangle

substitute

8=b^2

b=2\sqrt{2}\ units

therefore

The lengths of the sides of the square base are  2\sqrt{2}\ units

3 0
3 years ago
What do you get when you round to the nearest hundredth? HELPPP ME
KatRina [158]

what number though I can probably help you but what number do you need to round.

8 0
3 years ago
Find the surface area of a rectangular prism that has the following dimensions.
Shalnov [3]
2(wl+hl+hw)=2·(14·2+11·2+11·14)=408
6 0
3 years ago
Read 2 more answers
An Equation is shown.<br><br> 8/10 + ___ = 93/100<br><br> What is the missing fraction?
s2008m [1.1K]
The missing fraction is 13/100
3 0
3 years ago
Read 2 more answers
Other questions:
  • Select the following series as arithmetic, geometric, or neither. 3/2 + 3/4 + 3/8 + 3/16 + 3/32
    15·1 answer
  • Find the endpoint of the line segment with the given endpoint and midpoint.
    15·1 answer
  • Can someone please help ASAP PLEASE!!!!!
    7·1 answer
  • What is the simplified form of the equation fraction 4 over 5 n minus fraction 3 over 5 equals fraction 1 over 5 n?
    9·2 answers
  • ?-66= 25 what is the ?
    12·1 answer
  • cash apping whoever comments the right answer! Put 1 and 2 when you answer the question . and put your cash app name .
    11·2 answers
  • What is the equation of the line that passes through the point (-6, -2) and has a<br> slope of 1/6
    13·1 answer
  • What is x to this question? <br> -8 (x-1) = -32
    15·1 answer
  • Please help!<br><br> what is the slope of a line that is parallel to y= 3x+5?
    14·1 answer
  • A free kick is given in a soccer game where the player taking the kick is 14 yards from the left post. the goalie is standing on
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!