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3 years ago

7

Solve the system of equations:

2 answers:

yanalaym [24]3 years ago

5 0

Answer:

$x + y = 9 \\ x + 2x = 9 \\ 3x = 9 \\ x = \frac{9}{3} \\ x = 3$

pav-90 [236]3 years ago

3 0

The answer is y= 3 (:

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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 .

<h3>Learn More:-</h3>

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Which function can be used to model the graphed geometric sequence?

andreev551 [17]

Answer: The correct option is 1, i.e., $f(x+1)=\frac{5}{6}f(x)$ .

Explanation:

The geometric sequence is in the form of,

$a,ar,ar^2,ar^3,....$

Where, a is the first term of the sequence and r is the common ratio of the sequence.

It means the $n_{th}$ term of the sequence is defined as,

$a_n=ar^{n-1}$

So the the $(n+1)_{th}$ term of the sequence is defined as,

$a_{n+1}=ar^n$

$a_{n+1}=r(ar^{n-1})$

$a_{n+1}=ra_n$

It means the geometric sequence is in the form of,

$f(x+1)=rf(x)$

Where, r be any constant.

From the options only $f(x+1)=\frac{5}{6}f(x)$ is in the form of $f(x+1)=rf(x)$ with common ratio $\frac{5}{6}$ .

Therefore, the function can be used to model the graphed geometric sequence is $f(x+1)=\frac{5}{6}f(x)$ .

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4 years ago

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On average, a server is tipped 95 cents for each customer served. If 18 customers are served, how much money, in dollars, should

OlgaM077 [116]

Answer:

it would be 17.10 because .95 x 18 is 17.1 and 17.1 and 17.10 are the same

Step-by-step explanation:

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3 years ago

Find the difference between 7/10 and 14/15 .

sasho [114]

Answer:

Step-by-step explanation:

7/10 is 70% to 1 and 14/15 is 93% to 1

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How does a nuclear power plant produce electricity?

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I believe the answer is A.Quickly moving neutrons coming out of the reaction are slowed down by water. The water heats up and turns into steam. The steam turns the turbine and produces electricity.

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3 years ago