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lilavasa [31]
3 years ago
8

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2

- n + 2
Mathematics
1 answer:
Lady_Fox [76]3 years ago
3 0
Let n = 1
then f(1) = 1^1 - 1 + 2 = 2   so it is true for n = 1
for the next number  after n   ( n+1) we have  f(n+1) =
 (n+1)^2  - (n+1) + 2 

= n^2 + 2n + 1 - n - 1 + 2 

= n^2 + n + 2 

= n(n+1) + 2

Now n(n+1) must be divisible by 2  because  either n is odd and n+1 is even OR n is even and n+1 is odd   and odd & even always = an even number.

So the function is divisible by 2  for n+1   We have shown that its true for n = 1 Therefore it must be true for n = 1,2,3,4 ... 
True for all  positive integers


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Penelope has $1,459.75 in her bank account. To pay her bills, she writes 4 checks for $200.25, $359.45, $125, and $299.35. Then
sammy [17]
1,459.75 - 200.25 - 359.45 - 125.00 - 299.35 = 475.70

475.70 + 375.00 = 850.70

Her account balance after the bills and deposit is $850.70
8 0
2 years ago
Read 2 more answers
The Michigan veteran population in 1990 was 14.2% and in 2000 was 12.4%.
Mandarinka [93]

Let x- intercept represents time

Let y-intercept represents population

Let 1990 represent initial year

Points are (0,14.2)(10,12.4)

slope m= \frac{y2-y1}{x2-x1}

= \frac{12.4-14.2}{10-0} =\frac{-1.8}{10} = -0.08

General linear equation

y= mx+b --------------(i)

here m is slope and b is intercept

plug the 1st point in equation

14.2= -0.018(0) + b

b=14.2

y= 0.018x + 14.2

replace y with p(t) and x with t

p(t) = -0.018t + 14.2


6 0
2 years ago
PLEASE HELP!!!! 20 POINTS!!!!
galben [10]
These are two questions and two answers.

Question 1) Which of the following polar equations is equivalent to the parametric equations below? 

<span> x=t²
y=2t</span>

Answer: option <span>A.) r = 4cot(theta)csc(theta)
</span>

Explanation:

1) Polar coordinates ⇒ x = r cosθ and y = r sinθ

2) replace x and y in the parametric equations:

r cosθ = t²
r sinθ = 2t

3) work r sinθ = 2t

r sinθ/2 = t 
(r sinθ / 2)² = t²

4) equal both expressions for t²

r cos θ = (r sin θ / 2 )²

5) simplify

r cos θ = r² (sin θ)² / 4

4 = r (sinθ)² / cos θ

r = 4 cosθ / (sinθ)²

r = 4 cot θ csc θ ↔ which is the option A.


Question 2) Which polar equation is equivalent to the parametric equations below?

<span> x=sin(theta)cos(theta)+cos(theta)
y=sin^2(theta)+sin(theta)</span>

Answer: option B) r = sinθ + 1


Explanation:

1) Polar coordinates ⇒ x = r cosθ, and y = r sinθ

2) replace x and y in the parametric equations:

a) r cosθ = sin(θ)cos(θ)+cos(θ)
<span> b) r sinθ =sin²(θ)+sin(θ)</span>

3) work both equations

a) r cosθ = sin(θ)cos(θ)+cos(θ) ⇒ r cosθ = cosθ [ sin θ + 1]  ⇒ r = sinθ + 1


<span> b) r sinθ =sin²(θ)+sin(θ) ⇒ r sinθ = sinθ [sinθ + 1] ⇒ r = sinθ + 1
</span><span>
</span><span>
</span>Therefore, the answer is r = sinθ + 1 which is the option B.
6 0
2 years ago
Read 2 more answers
Make r the subject of the formula in I=E/R+r​
allochka39001 [22]

Step-by-step explanation:

I =  \frac{E }{ R}  + r

I  -  \frac{ E }{ R}  = r

\frac{I R -E  }{ R}  = r

HOPE IT HELPS Y

8 0
2 years ago
What is 10 less than 14 in algebraic expression?
Cerrena [4.2K]

14-10=? less than indicates that a minus sign is needed.

7 0
3 years ago
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