Answer:
1 7/8
Step-by-step explanation:
The first step is to find the least common denominator of 3/8 and 9/4, or the least common multiple of 8 and 4. LCD is 8. Make the denominators the same as the LCD.
-3/8 + 9/4x2
-3/8 + 18/8
-3+18/8
15/8
1 7/8
Hope this helps!:) Best of luck!
Complete Question
Find a formula for the sum of n terms. ![\sum\limits_{i=1}^n ( 8 + \frac{i}{n} )(\frac{2}{n} )](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bi%3D1%7D%5En%20%20%28%208%20%2B%20%5Cfrac%7Bi%7D%7Bn%7D%20%29%28%5Cfrac%7B2%7D%7Bn%7D%20%29)
Use the formula to find the limit as ![n \to \infty](https://tex.z-dn.net/?f=n%20%5Cto%20%5Cinfty)
Answer:
![K_n = \frac{n + 73 }{n}](https://tex.z-dn.net/?f=K_n%20%20%3D%20%20%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D)
![\lim_{n \to \infty} K_n = 1](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20K_n%20%20%3D%20%201)
Step-by-step explanation:
So let assume that
![K_n = \sum\limits_{i=1}^n ( 8 + \frac{i}{n} )(\frac{2}{n} )](https://tex.z-dn.net/?f=K_n%20%20%3D%20%20%5Csum%5Climits_%7Bi%3D1%7D%5En%20%20%28%208%20%2B%20%5Cfrac%7Bi%7D%7Bn%7D%20%29%28%5Cfrac%7B2%7D%7Bn%7D%20%29)
=> ![K_n = \sum\limits_{i=1}^n ( \frac{16}{n} + \frac{2i}{n^2} )](https://tex.z-dn.net/?f=K_n%20%20%3D%20%20%5Csum%5Climits_%7Bi%3D1%7D%5En%20%20%28%20%5Cfrac%7B16%7D%7Bn%7D%20%2B%20%5Cfrac%7B2i%7D%7Bn%5E2%7D%20%29)
=> ![K_n = \frac{2}{n} \sum\limits_{i=1}^n (8) + \frac{2}{n^2} \sum\limits_{i=1}^n(i)](https://tex.z-dn.net/?f=K_n%20%20%3D%20%5Cfrac%7B2%7D%7Bn%7D%20%20%5Csum%5Climits_%7Bi%3D1%7D%5En%20%288%29%20%2B%20%5Cfrac%7B2%7D%7Bn%5E2%7D%20%20%20%5Csum%5Climits_%7Bi%3D1%7D%5En%28i%29)
Generally
![\sum\limits_{i=1}^n (k) = \frac{1}{2} n (n + 1)](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bi%3D1%7D%5En%20%28k%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20n%20%20%28n%20%2B%201%29)
So
![\sum\limits_{i=1}^n (8) = \frac{1}{2} * 8* (8 + 1)](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bi%3D1%7D%5En%20%288%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20%2A%208%2A%20%20%288%20%2B%201%29)
![\sum\limits_{i=1}^n (8) = 36](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bi%3D1%7D%5En%20%288%29%20%3D%2036)
and
![\sum\limits_{i=1}^n (i) = \frac{1}{2} n (n + 1)](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bi%3D1%7D%5En%20%28i%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%20n%20%20%28n%20%2B%201%29)
Therefore
![K_n = \frac{72}{n} + \frac{2}{n^2} * \frac{1}{2} n (n + 1 )](https://tex.z-dn.net/?f=K_n%20%20%3D%20%5Cfrac%7B72%7D%7Bn%7D%20%2B%20%5Cfrac%7B2%7D%7Bn%5E2%7D%20%20%20%2A%20%20%5Cfrac%7B1%7D%7B2%7D%20%20n%20%28n%20%2B%201%20%29)
![K_n = \frac{72}{n} + \frac{1}{n} (n + 1 )](https://tex.z-dn.net/?f=K_n%20%20%3D%20%5Cfrac%7B72%7D%7Bn%7D%20%2B%20%20%20%20%5Cfrac%7B1%7D%7Bn%7D%20%20%20%28n%20%2B%201%20%29)
![K_n = \frac{72}{n} + 1 + \frac{1}{n}](https://tex.z-dn.net/?f=K_n%20%20%3D%20%5Cfrac%7B72%7D%7Bn%7D%20%2B%20%20%201%20%2B%20%20%5Cfrac%7B1%7D%7Bn%7D)
![K_n = \frac{72 + 1 + n }{n}](https://tex.z-dn.net/?f=K_n%20%20%3D%20%20%5Cfrac%7B72%20%2B%20%201%20%2B%20%20n%20%7D%7Bn%7D)
![K_n = \frac{n + 73 }{n}](https://tex.z-dn.net/?f=K_n%20%20%3D%20%20%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D)
Now ![\lim_{n \to \infty} K_n = \lim_{n \to \infty} [\frac{n + 73 }{n} ]](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20K_n%20%20%3D%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D%20%5D)
=> ![\lim_{n \to \infty} [\frac{n + 73 }{n} ] = \lim_{n \to \infty} [\frac{n}{n} + \frac{73 }{n} ]](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D%20%5D%20%20%3D%20%20%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%7D%7Bn%7D%20%20%2B%20%20%5Cfrac%7B73%20%7D%7Bn%7D%20%20%5D)
=> ![\lim_{n \to \infty} [\frac{n + 73 }{n} ] = \lim_{n \to \infty} [1 + \frac{73 }{n} ]](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D%20%5D%20%20%3D%20%20%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B1%20%2B%20%20%5Cfrac%7B73%20%7D%7Bn%7D%20%20%5D)
=> ![\lim_{n \to \infty} [\frac{n + 73 }{n} ] = \lim_{n \to \infty} [1 ] + \lim_{n \to \infty} [\frac{73 }{n} ]](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D%20%5D%20%20%3D%20%20%20%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B1%20%5D%20%2B%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5B%5Cfrac%7B73%20%7D%7Bn%7D%20%20%5D)
=> ![\lim_{n \to \infty} [\frac{n + 73 }{n} ] = 1 + 0](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5B%5Cfrac%7Bn%20%2B%2073%20%7D%7Bn%7D%20%5D%20%20%3D%20%201%20%20%2B%20%200)
Therefore
![\lim_{n \to \infty} K_n = 1](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20K_n%20%20%3D%20%201)
Answer:
5579183747482837371828282
<h2>
The greatest possible value of the second root, β = 54 </h2>
Step-by-step explanation:
The given quadratic equation:
![x^2-5mx+6m^2=0](https://tex.z-dn.net/?f=x%5E2-5mx%2B6m%5E2%3D0)
Let α and β be the roots of the given quadratic equation.
α = 36
To find, the greatest possible value of the second root ( β) = ?
∴ The sum of the roots,
α + β = ![\dfrac{-b}{a}](https://tex.z-dn.net/?f=%5Cdfrac%7B-b%7D%7Ba%7D)
⇒ 36 + β =
⇒ 5m = 36 + β ............. (1)
The product of the roots,
α.β = ![\dfrac{c}{a}](https://tex.z-dn.net/?f=%5Cdfrac%7Bc%7D%7Ba%7D)
⇒ ![36.\beta=\dfrac{6m^2}{1}](https://tex.z-dn.net/?f=36.%5Cbeta%3D%5Cdfrac%7B6m%5E2%7D%7B1%7D)
⇒
............. (2)
From equations (1) and (2), we get
⇒ ![(\dfrac{36+\beta}{5})^{2}=6\beta](https://tex.z-dn.net/?f=%28%5Cdfrac%7B36%2B%5Cbeta%7D%7B5%7D%29%5E%7B2%7D%3D6%5Cbeta)
⇒ ![\beta^2-78\beta+1296=0](https://tex.z-dn.net/?f=%5Cbeta%5E2-78%5Cbeta%2B1296%3D0)
⇒ ![\beta^2-54B\beta-24B\beta+1296=0](https://tex.z-dn.net/?f=%5Cbeta%5E2-54B%5Cbeta-24B%5Cbeta%2B1296%3D0)
⇒ β(β - 54) - 24(β - 54) = 0
⇒ (β - 54)(β - 24) = 0
⇒ β - 54 = 0 or, β - 24 = 0
⇒ β = 54 or, β = 24
∴ The greatest possible value of the second root, β = 54
The answer is 1/2 and 7/8.