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3 years ago

What fraction is between 3/4 and 1 ?

Mathematics

1 answer:

VLD [36.1K]3 years ago

8 0

7/8 is between 3/4 and 1

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Use induction to prove: For every integer n > 1, the number n5 - n is a multiple of 5.

nignag [31]

Answer:

we need to prove : for every integer n>1, the number $n^{5}-n$ is a multiple of 5.

1) check divisibility for n=1, $f(1)=(1)^{5}-1=0$ (divisible)

2) Assume that $f(k)$ is divisible by 5, $f(k)=(k)^{5}-k$

3) Induction,

$f(k+1)=(k+1)^{5}-(k+1)$

$=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1$

$=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k$

Now, $f(k+1)-f(k)$

$f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)$

$f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k$

$f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k$

Take out the common factor,

$f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)$ (divisible by 5)

add both the sides by f(k)

$f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)$

We have proved that difference between $f(k+1)$ and $f(k)$ is divisible by 5.

so, our assumption in step 2 is correct.

Since $f(k)$ is divisible by 5, then $f(k+1)$ must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number $n^{5}-n$ is a multiple of 5.

3 0

3 years ago

Help Me PLEASE!!! Simplify The Equation (2nd question)

scoray [572]

I think it would be 8-3, not 7. 8x - 2x + 5x = (8-5)x + 5x

5 0

3 years ago

What is the slope of the line that passes through (-4, 5) and (1, 3)?

Tasya [4]

M=-2/5 is the answer to this equation

7 0

3 years ago

Multiply and and simplify : (a+b)^2<br> Pls help easy

Lena [83]

Answer:a^2+2ab+b^2

Step-by-step explanation:

First write it out in extended form, (a+b)(a+b) then multiply each term by the other in each parenthesis, such as a*a+b*b+a*b+a*b

Then, once you have done that, simplify it. you will then get the answer a^2+2ab+b^2

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3 years ago

Samantha works at a nursery where she pots plants. It takes her 112 hours to pot 12 plants. Question 1 Part A How many plants ca

PolarNik [594]

Answer:

Step-by-step explanation:

112/12= 9.3

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3 years ago