Use mathematical induction to prove the statement is true for all positive integers n. The integer n3 + 2n is divisible by 3 for every positive integer n.

Answers

Answer 1

Answer: 1. prove it is true for n=1
2. assume n=k
3. prove that n=k+1 is true as well

so

1.
$(n^3+2n)/(3)$ =
$(1^3+2(1))/(3)$ =
$(1+2)/(3)$ =1
we got a whole number, true

2.
$(k^3+2k)/(3)$
if everything clears, then it is divisble

3.
$((k+1)^3+2(k+1))/(3)$ =
$((k+1)^3+2(k+1))/(3)$ =
$(k^3+3k^2+3k+1+2k+2))/(3)$ =
$(k^3+3k^2+5k+3))/(3)$
we know that if z is divisble by 3, then z+3 is divisble b 3
also, 3k/3=a whole number when k= a whole number

$(k^3+2k)/(3) + (3k^2+3k+3)/(3)$ =
$(k^3+2k)/(3) + k^2+k+1$ =
since the k²+k+1 part cleared, it is divisble by 3

we found that it simplified back to $(k^3+2k)/(3)$

done

Answer 2

Answer:

Answer:

We have to use the mathematical induction to prove the statement is true for all positive integers n.

The integer $n^3+2n$ is divisible by 3 for every positive integer n.

for n=1

$n^3+2n=1+2=3$ is divisible by 3.

Hence, the statement holds true for n=1.

Let us assume that the statement holds true for n=k.

i.e. $k^3+2k$ is divisible by 3.---------(2)

Now we will prove that the statement is true for n=k+1.

i.e. $(k+1)^3+2(k+1)$ is divisible by 3.

We know that:

$(k+1)^3=k^3+1+3k^2+3k$

and $2(k+1)=2k+2$

Hence,

$(k+1)^3+2(k+1)=k^3+1+3k^2+3k+2k+2\n\n(k+1)^3+2(k+1)=(k^3+2k)+3k^2+3k+3=(k^3+2k)+3(k^2+k+1)$

As we know that:

$(k^3+2k)$ was divisible as by using the second statement.

Also:

$3(k^2+k+1)$ is divisible by 3.

Hence, the addition:

$(k^3+2k)+3(k^2+k+1)$ is divisible by 3.

Hence, the statement holds true for n=k+1.

Hence by the mathematical induction it is proved that:

The integer $n^3+2n$ is divisible by 3 for every positive integer n.

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Two black chips and three red chips are put into a bag. Two points are awarded for each black chip drawn, and one point is lost for each red chip drawn. What is the expected value for each round if there are two draws per round and the chips are replaced after each draw?

Answers

The answer is 0.4.

The values are:
- for the black chip : x₁ = 2
- for the red chip: x₂ = -1

Let's first calculate the possibilities of each chip. There are in total 5 chips (two black chips and three red chips) in the bag
- the possibility to draw the black chip is 2 out of 5: P₁ = 2/5
- the possibility to draw the red chip is 3 out of 5: P₂ = 3/5

In this example we have 4 different events:
1. Drawing of two black chips: P₃ = P₁ · P₁ = 2/5 · 2/5 = 4/25
2. Drawing of one black chip and then one red chip: P₄ = P₁ · P₂ = 2/5 · 3/5 = 6/25
3. Drawing of one red chip and then one black chip: P₅ = P₂ · P₁ = 3/5 · 2/5 = 6/25
5. Drawing of two black chips: P₆ = P₂ · P₂ = 3/5 · 3/5 = 9/25

Therefore, the expected value for each round if there are two draws per round and the chips are replaced after each draw is 0.4:
P = (x₁ + x₁) · P₃ + (x₁ + x₂) · P₄ + (x₁ + x₂) · P₅ + (x₂ + x₂) · P₆
P = (2+2) · 4/25 + (2-1) · 6/25 + (2-1) · 6/25 + (-1 + -1) · 9/25
P = 4 · 4/25 + 1 · 6/25 + 1 · 6/25 + -2 · 9/25
P = 16/25 + 6/25 + 6/25 - 18/25
P = 28/25 - 18/25
P = 10/25 = 0.4

Answer:

0.4 is correct

Step-by-step explanation:

Evaluate the following expression using the values given:Find -3a2 - b3 + 3c2 - 2b3 if a = 2, b = -1, and c = 3.

Answers

-3(2^2) - 3(-1) + 3(3^2) - 2(-1^3)
-3(4) - 3(-1) + 3(9) - 2(-1)
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in two complementary angles, the measure of one angle is 6 more than twice the measure of the other. the measures of these two angles are

Answers

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When a bill is not paid by the due date, a late fee is added each month until the bill is paid. The values in the table represent the total amount due, in dollars, for each month that a payment remains overdue.Months | Total due
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Answers

The total amount due, in dollars, is an illustration of a linear function

The equation of the function is $y = 25x + 250$

Represent the month with x, and the total due with y.

Start by calculating the slope (m) using:

$m = (y_2 -y_1)/(x_2 -x_1)$

So, we have:

$m = (375 - 300)/(5 - 2)$

Simplify

$m = (75)/(3)$

Divide 75 by 3

$m = 25$

The linear equation is then calculated as:

$y = m(x - x_1) + y_1$

So, we have:

$y = 25(x - 2) + 300$

Open bracket

$y = 25x - 50 + 300$

Add 300 to -50

$y = 25x + 250$

Hence, the equation of the function is $y = 25x + 250$

Read more about linear equations at:

brainly.com/question/4074386

Find the area bounded by the curve y=x2 and the straight line y=2+xA.4 1/6
B. 4 1/2
C. 5 1/6
D. 5 1/2

Answers

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Find the geometric mean x of each pair of numbers. 1.5 and 84

Answers

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