Answer:
X = {8n – 7n – 1 : n ϵ N}
Y = {49 (n –1) : n ϵ N} — (1) [all terms divisible by 49]
X = {8n – 7n – 1}
= (1 + 7)n – 7n – 1
= 1 + nC1 7 + nC2 72 + ….. + nCn 7n – 7n – 1
= nC2 72 + …… + nCn 7n
= 49 [nC2 + nC3 7 + …. nCn 7n-2] — (2)
From (1) and (2),
X is divisible by 49.
Y has all multiples of 49.
X ⊂ Y
We have: ![f(x)=x^2+6x+17](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2%2B6x%2B17)
Method 1.
Use: ![(*)\ \ \ \ \ (a+b)^2=a^2+2ab+b^2](https://tex.z-dn.net/?f=%28%2A%29%5C%20%5C%20%5C%20%5C%20%5C%20%28a%2Bb%29%5E2%3Da%5E2%2B2ab%2Bb%5E2)
and the vertex form ![f(x)=a(x-h)^2+k](https://tex.z-dn.net/?f=f%28x%29%3Da%28x-h%29%5E2%2Bk)
(h, k) - the coordinates of a vertex
![x^2+6x+17=x^2+2\cdot x\cdot3+17=\underbrace{x^2+2\cdot x\cdot3+3^2}_{(*)}-3^2+17\\\\=(x+3)^2+8\to h=-3;\ k=8](https://tex.z-dn.net/?f=x%5E2%2B6x%2B17%3Dx%5E2%2B2%5Ccdot%20x%5Ccdot3%2B17%3D%5Cunderbrace%7Bx%5E2%2B2%5Ccdot%20x%5Ccdot3%2B3%5E2%7D_%7B%28%2A%29%7D-3%5E2%2B17%5C%5C%5C%5C%3D%28x%2B3%29%5E2%2B8%5Cto%20h%3D-3%3B%5C%20k%3D8)
Answer: (-3, 8)
Method 2.
For
the coordinates of a vertex (h, k) are equal
![h=\dfrac{-b}{2a};\ k=f(h)=\dfrac{-(b^2-4ac)}{4a}](https://tex.z-dn.net/?f=h%3D%5Cdfrac%7B-b%7D%7B2a%7D%3B%5C%20k%3Df%28h%29%3D%5Cdfrac%7B-%28b%5E2-4ac%29%7D%7B4a%7D)
We have: ![f(x)=x^2+6x+17\to a=1;\ b=6;\ c=17](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2%2B6x%2B17%5Cto%20a%3D1%3B%5C%20b%3D6%3B%5C%20c%3D17)
Substitute:
![h=\dfrac{-6}{2\cdot1}=\dfrac{-6}{2}=-3\\\\k=f(-3)=(-3)^2+6(-3)+17=9-18+17=8](https://tex.z-dn.net/?f=h%3D%5Cdfrac%7B-6%7D%7B2%5Ccdot1%7D%3D%5Cdfrac%7B-6%7D%7B2%7D%3D-3%5C%5C%5C%5Ck%3Df%28-3%29%3D%28-3%29%5E2%2B6%28-3%29%2B17%3D9-18%2B17%3D8)
Answer: (-3, 8)
Answer:
The constant of proportionality is 5.
Step-by-step explanation:
If the quantity q varies inversely with the square of m and directly with the product of r and x, then the variation equation is
![q=k\frac{rx}{m^2}](https://tex.z-dn.net/?f=q%3Dk%5Cfrac%7Brx%7D%7Bm%5E2%7D)
When q=2.5, m= 4 and the
product of rx = 8.
We plug in this values to get;
![2.5=k(\frac{8}{4^2})](https://tex.z-dn.net/?f=2.5%3Dk%28%5Cfrac%7B8%7D%7B4%5E2%7D%29)
![2.5=k(\frac{8}{16})](https://tex.z-dn.net/?f=2.5%3Dk%28%5Cfrac%7B8%7D%7B16%7D%29)
![2.5=\frac{1}{2}k](https://tex.z-dn.net/?f=2.5%3D%5Cfrac%7B1%7D%7B2%7Dk)
Multiply both sides by 2
![k=2.5\times 2=5](https://tex.z-dn.net/?f=k%3D2.5%5Ctimes%202%3D5)
The constant of proportionality is 5.
Answer:
m=2
Step-by-step explanation:
The slope-intercept form is y=mx+b, where m is the slope and b is the y-intercept.
y=mx+b
Subtract 4x from both sides of the equation.
−2y=5−4x
Divide each term by −2 and simplify.
y=2x−5/2
Answer:
158.428571429
158 remainder 6
Step-by-step explanation: