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lana [24]
3 years ago
8

Find the infinite sum of 10,5,2.5,1.25...

Mathematics
2 answers:
scZoUnD [109]3 years ago
8 0
To the risk of sounding redundant.

notice it goes from 10, to 5, to 2.5 and so forth, the next term is half the previous one, namely the multiplier, or "common ratio" is 1/2.

10 * 1/2 is 5, 5 * 1/2 is 2.5 and so on.

now, the Sum is going to infinity, however, when the common ratio is a fraction less than 1 and more than 0, either negative or positive, it is convergent, namely it reaches a limit, it doesn't go to neverland.

so, in this case, the common ratio is indeed | r | < 1, and therefore, it does reach a limit, keeping in mind the first term's value is 10.

\bf \textit{Sum of an infinite sequence}\\\\&#10;S=\sum\limits_{i=0}^\infty~a_1\cdot r^i,\quad |~r~|~
____ [38]3 years ago
3 0
Notice this is a geometric progression since each number multiplied by some factor equals the next number in the sequence, in this case,

10 \times r = 5 \Rightarrow r=0.5

Then by applying the formula for sum to infinity of a geometric progression,
S_\infty = \frac{a}{1-r} = \frac{10}{1-0.5} = 20
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write an equation for the perpendicular bisector of the line joining the two points. PLEASE do 4,5 and 6
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Answer:

4. The equation of the perpendicular bisector is y = \frac{3}{4} x - \frac{1}{8}

5. The equation of the perpendicular bisector is y = - 2x + 16

6. The equation of the perpendicular bisector is y = -\frac{3}{2} x + \frac{7}{2}

Step-by-step explanation:

Lets revise some important rules

  • The product of the slopes of the perpendicular lines is -1, that means if the slope of one of them is m, then the slope of the other is -\frac{1}{m} (reciprocal m and change its sign)
  • The perpendicular bisector of a line means another line perpendicular to it and intersect it in its mid-point
  • The formula of the slope of a line is m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}
  • The mid point of a segment whose end points are (x_{1},y_{1}) and (x_{2},y_{2}) is (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})
  • The slope-intercept form of the linear equation is y = m x + b, where m is the slope and b is the y-intercept

4.

∵ The line passes through (7 , 2) and (4 , 6)

- Use the formula of the slope to find its slope

∵ x_{1} = 7 and x_{2} = 4

∵ y_{1} = 2 and y_{2} = 6

∴ m=\frac{6-2}{4-7}=\frac{4}{-3}

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line = \frac{3}{4}

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point = (\frac{7+4}{2},\frac{2+6}{2})

∴ The mid-point = (\frac{11}{2},\frac{8}{2})=(\frac{11}{2},4)

- Substitute the value of the slope in the form of the equation

∵ y = \frac{3}{4} x + b

- To find b substitute x and y in the equation by the coordinates

   of the mid-point

∵ 4 = \frac{3}{4} × \frac{11}{2} + b

∴ 4 = \frac{33}{8} + b

- Subtract  \frac{33}{8} from both sides

∴ -\frac{1}{8} = b

∴ y = \frac{3}{4} x - \frac{1}{8}

∴ The equation of the perpendicular bisector is y = \frac{3}{4} x - \frac{1}{8}

5.

∵ The line passes through (8 , 5) and (4 , 3)

- Use the formula of the slope to find its slope

∵ x_{1} = 8 and x_{2} = 4

∵ y_{1} = 5 and y_{2} = 3

∴ m=\frac{3-5}{4-8}=\frac{-2}{-4}=\frac{1}{2}

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line = -2

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point = (\frac{8+4}{2},\frac{5+3}{2})

∴ The mid-point = (\frac{12}{2},\frac{8}{2})

∴ The mid-point = (6 , 4)

- Substitute the value of the slope in the form of the equation

∵ y = - 2x + b

- To find b substitute x and y in the equation by the coordinates

   of the mid-point

∵ 4 = -2 × 6 + b

∴ 4 = -12 + b

- Add 12 to both sides

∴ 16 = b

∴ y = - 2x + 16

∴ The equation of the perpendicular bisector is y = - 2x + 16

6.

∵ The line passes through (6 , 1) and (0 , -3)

- Use the formula of the slope to find its slope

∵ x_{1} = 6 and x_{2} = 0

∵ y_{1} = 1 and y_{2} = -3

∴ m=\frac{-3-1}{0-6}=\frac{-4}{-6}=\frac{2}{3}

- Reciprocal it and change its sign to find the slope of the ⊥ line

∴ The slope of the perpendicular line = -\frac{3}{2}

- Use the rule of the mid-point to find the mid-point of the line

∴ The mid-point = (\frac{6+0}{2},\frac{1+-3}{2})

∴ The mid-point = (\frac{6}{2},\frac{-2}{2})

∴ The mid-point = (3 , -1)

- Substitute the value of the slope in the form of the equation

∵ y = -\frac{3}{2} x + b

- To find b substitute x and y in the equation by the coordinates

   of the mid-point

∵ -1 = -\frac{3}{2} × 3 + b

∴ -1 = -\frac{9}{2} + b

- Add  \frac{9}{2}  to both sides

∴ \frac{7}{2} = b

∴ y = -\frac{3}{2} x + \frac{7}{2}

∴ The equation of the perpendicular bisector is y = -\frac{3}{2} x + \frac{7}{2}

8 0
3 years ago
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