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4 years ago

Can you help me solve 12 through 16 I don't get it

Mathematics

2 answers:

Alja [10]4 years ago

7 0

Can you take closer picture please:)

morpeh [17]4 years ago

3 0

All you have to do is divide
the fractions in the problems and get your answer for example number 15 4/5÷1/4
4×1=4
5×4=20 = 4/20 miles to go

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Prove that 1³+2³+....n³=n²(n+1)²/4. principle of mathematics induction

alex41 [277]

<h3>The Simplified Question:-</h3>

$\sf 1^3+2^3\dots n^3=\dfrac{n^2(n+1)^2}{2}$

$\\ \sf\longmapsto 1^3+2^3+\dots n^3=\left(\dfrac{n(n+1)}{2}\right)^2$

<h3>Solution:-</h3>

Let

$\\ \sf\longmapsto P(n)=1^3+2^3\dots n^3=\left(\dfrac{n(n+1)}{2}\right)^2$

For n=1

$\\ \sf\longmapsto P(1)=\left(\dfrac{1(1+1)}{2}\right)^2$

$\\ \sf\longmapsto P(1)=\left(\dfrac{1(2)}{2}\right)^2$

$\\ \sf\longmapsto P(1)=\left(\dfrac{2}{2}\right)^2$

$\\ \sf\longmapsto P(1)=(1)^2$

$\\ \bf\longmapsto P(1)=1=1^3$

Let k be any positive integer.

$\\ \sf\longmapsto P(k)= 1^3+2^3\dots k^3=\left(\dfrac{k(k+1)}{2}\right)^2$

We have to prove that p(k+1) is true.

consider

$\sf 1^3+2^3\dots k^3+(k+1)^3$

$\\ \sf\longmapsto \left(\dfrac{k(k+1)}{2}\right)^2+(k+1)^3$

$\\ \sf\longmapsto \dfrac{k^2(k+1)^2}{4}+(k+1)^3$

$\\ \sf\longmapsto \dfrac{k^2(k+1)^2+4(k+1)^3}{4}$

$\\ \sf\longmapsto \dfrac{k+1)^2\left\{k^2+4k+4\right\}}{4}$

$\\ \sf\longmapsto \dfrac{(k+1)^2(k+2)^2}{4}$

$\\ \sf\longmapsto \dfrac{(k+1)^2(k+1+1)^2}{4}$

$\\ \sf\longmapsto \left(\dfrac{(k+1)(k+1+1)}{2}\right)^2$

$\\ \sf\longmapsto (1^3+2^3+3^3\dots k^3)+(k+1)^3$

Thus P(k+1) is true whenever P(k) is true.

Hence by the Principal of mathematical induction statement P(n) is true for $\bf n\epsilon N.$

Note:-

We can solve without simplifying the Question .I did it for clear steps and understanding .

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3 years ago

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Evaluate the function for<br> f(x) = x + 3 and g(x) = x2 − 2.<br> (f − g)(0)<br><br> (f − g)(0) =

adelina 88 [10]

Hello,

(f-g)(0)=f(0)-g(0)=3-(-2)=5

8 0

3 years ago

Greatest Common Factor and Factoring by Grouping. Please solve this maths

andrey2020 [161]

The greatest common factor and factoring by grouping of the following is;

3m
2a
8xy
5mn
3(r + t)²
z²(m + n)⁴

<h3>Greatest common factor</h3>

The greatest common factor is the largest positive integer or polynomial that is a divisor of several different numbers. For instance, the greatest common divisor of 66, 30 and 18 is 6.

9m³, 3m², 15m

= 3m

4a², 6a, 2a³

= 2a

16xy³, 24x²y², 8x²y

= 8xy

10m²n², 25mn³, 50m²n

= 5mn

6m(r + t)², 3p(r + t)⁴

= 3(r + t)²

7z²(m + n)⁴, 9z³(m + n)^5

= z²(m + n)⁴

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2 years ago

Trisha needs 4.5 yards of fabric to do a project. If she already has 2/3 of a yard of fabric, what is the minimum amount of fabr

DochEvi [55]

1 yard = 3 feet
2/3 of a yard would be 2 feet

4.5 yards = 13.5 feet
13.5-2 = 11.5 feet

11.5 feet = 3.83 yards

therefore at least 11.5 feet or 3.83 yards must be purchased

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3 years ago

This is part two of the questions

vodomira [7]

Answer:

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