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
Naya [18.7K]
3 years ago
5

Prove that1/1+root 2 + 1/root2 + root3 + 1/root3 + root4 + 1/root8 +root9 = 2​

Mathematics
1 answer:
daser333 [38]3 years ago
4 0

Step-by-step explanation:

LHS:

\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots \ldots \ldots \ldots \ldots+\frac{1}{\sqrt{8}+\sqrt{9}}1+21+2+31+3+41+……………+8+91

Rationalizing the denominator, we get

\Rightarrow\left(\frac{1}{1+\sqrt{2}} \times \frac{1-\sqrt{2}}{1-\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}+\sqrt{3}} \times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}\right)+\left(\frac{1}{\sqrt{3}+\sqrt{4}} \times \frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}}\right)+\cdots \ldots+\left(\frac{1}{\sqrt{8}+\sqrt{9}} \times \frac{\sqrt{8}-\sqrt{9}}{\sqrt{8}-\sqrt{9}}\right)⇒(1+21×1−21−2)+(2+31×2−32−3)+(3+41×3−43−4)+⋯…+(8+91×8−98−9)

We know that,

\left(a^{2}-b^{2}\right)=(a+b)(a-b)(a2−b2)=(a+b)(a−b)

Now, on substituting the formula, we get,

=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+\frac{\sqrt{3}-\sqrt{4}}{3-4}+\cdots \ldots \cdot \frac{(\sqrt{8}-\sqrt{9})}{8-9}=1−21−2+2−32−3+3−43−4+⋯…⋅8−9(8−9)

\Rightarrow \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{8}+\sqrt{9}}=(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+\cdots+(\sqrt{9}-\sqrt{8})⇒1+21+

You might be interested in
What are 2 real life slope problems?
Scilla [17]
Umm no you need to do your own work and do not be asking for answers
Nothing is handed to you ....you need to work HARD to get what you want.

4 0
3 years ago
Write the subtraction as addition , -9 subtracting (-4)
leva [86]
The answer would be -9 + -4
8 0
3 years ago
Read 2 more answers
The total revenue from the sale of a popular book is approximated by the rational function, R(x)=1100x^2/x^2 + 4 where x is the
vazorg [7]

Answer:

The answer is below

Step-by-step explanation:

The total revenue from the sale of a popular book is approximated by the rational R(x) = 1100x^2/x^2 + 4, where x is the number of years since publication and the total revenue in millions of dollars. Use this function to complete parts a through d. (a) Find the total revenue at the end of the first year. $ million (b) Find the total revenue at the end of the second year. $ million (c) Find the domain of function R. Choose the correct domain below. {x | x is a real number and x greaterthanorequalto 0} {x | x is a real number and x lessthanorequalto 5} {x | x is a real number and x notequalto 2, x notequalto 5} {x | x is a real number and x notequalto 2}

Solution:

Revenue is the total money made from the selling of a product or item.

a) Given the revenue function:

R(x)=\frac{1100x^2}{x^2+4}

At the end of the first year (i.e. x = 1), the venue is given as:

R(1)=\frac{1100(1)^2}{(1)^2+4} =220\\\\R(1)=\$ 220\ million

b) At the end of the second year (i.e. x = 2), the venue is given as:

R(2)=\frac{1100(2)^2}{(2)^2+4} =220\\\\R(2)=\$ 550\ million

c) The domain of the function is the value of the denominator for which the value of the denominator is not zero

x² + 4 ≠ 0

The domain is {x| x is a real number and x ≥ 0}

3 0
3 years ago
What is 39,800,000 in scientific notation? A. 3.98 x 10 B. 398 x 10 C. 3.98 x 10 D. 39.8 x 10
Masteriza [31]

Answer:

3.98 x 10^7

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
Alex spent $23 on a bus ride. The bus charges $0.35 a mile plus $2 a trip. How
Svetlanka [38]
67.7 cus 23 divided by .35 is 65.7+2 =67.7
7 0
3 years ago
Other questions:
  • Consider the recursively defined function below. f(1)=-5.25 f(n)=f(n-1)+1.75, for n=2,3,4,.... Create the first five terms of th
    15·1 answer
  • Given a+b+c=7 and a^2+b^2+c^2=25 Find (a+b) ^2+(a+c) ^2+(b+c) ^2
    13·1 answer
  • 6÷2(1+2)<br> I WILL GIVE 15 PTS TO CORRECT PERSON
    6·1 answer
  • Which of the following is most likely the next step in the series?
    9·2 answers
  • How would I reduce 540/600 using prime factorization?
    10·2 answers
  • Three consecutive odd integers have a sum of 33.
    7·2 answers
  • Jake and Seth put some money into their money boxes every week. The amount of money (y), in dollars, in their money boxes after
    10·2 answers
  • Plz help due tomorrow
    13·2 answers
  • What is the prime factorization of -14x^4
    7·1 answer
  • John had $20. He earned $5, spent $10, earned $5 again, and then spent $3. After this series of earnings and expenses, how much
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!