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Over [174]
3 years ago
10

I have two questions. Please try to answer both!

Mathematics
1 answer:
OlgaM077 [116]3 years ago
7 0

Answer:

(a)59

(b)d=\dfrac{1}{2}

Step-by-step explanation:

<u>Part A</u>

The nth term of an arithmetic sequence, T_n=a+(n-1)d

Let the first angle of the triangle =a

Let the common difference = d.

Since the measures of the angles are in an arithmetic progression

We have:

a + (a + d) + (a + 2d) = 180

3a+3d=180

a+d=60

Since a or d cannot be equal to zero, the minimum possible values of a and d is 1 and the maximum possible values of a and d is 59.

Therefore, there are 59 non-similar triangles.

<u>Part B</u>

In an arithmetic sequence

a_7 -2a_4 = 1, a_3 = 0.

2a_4=a_7-1\\a_4=\dfrac{a_7-1}{2}

Common difference,

d=a_4-a_3\\=\dfrac{a_7-1}{2}-0\\d=\dfrac{a_7-1}{2}

a_7=a_4+3d\\=\dfrac{a_7-1}{2}+3(\dfrac{a_7-1}{2})\\=\dfrac{a_7-1}{2}+\dfrac{3a_7-3}{2}\\=\dfrac{a_7-1+3a_7-3}{2}\\a_7=\dfrac{4a_7-4}{2}\\$Cross multiply\\2a_7=4a_7-4\\2a_7-4a_7=-4\\-2a_7=-4\\a_7=2

Therefore:

d=\dfrac{2-1}{2}\\d=\dfrac{1}{2}

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Help please show work
Romashka [77]

Answer:

Just subtract a tuig from the thung

Step-by-step explanation:

Heres an example

subtract 7 from 12 an you will get 5 for number 1 do that for the rest

8 0
2 years ago
3,6,11,18,27,38
MakcuM [25]

Answer:  The pattern is x^2+2

where x is the term number

Example: the 5th term is 27 because x = 5 leads to x^2+2 = 5^2+2 = 27

=================================================

Explanation:

  • The jump from 3 to 6 is +3
  • The jump from 6 to 11 is +5
  • The jump from 11 to 18 is +7
  • The jump from 18 to 27 is +9
  • The jump from 27 to 38 is +11

The pattern of jumps is: 3, 5, 7, 9, 11

Those increments are going up by 2 each time.

Since we have a consistent pattern of increments, this means that the sequence follows a quadratic model.

Quadratics are stuff like x^2+7x+10 or 3x^2-7. The leading term has an exponent of 2.

-----------

If x is the term number and y is the term itself, then we have these points

(1,3)

(2,6)

(3,11)

(4,18)

(5,27)

(6,38)

The x coordinates increase by 1 each time. The y coordinates are the terms given by your teacher.

Pick exactly 3 of those points. I'll pick the first 3.

Why 3? Because we'll have 3 unknowns to solve for, in which we'll need 3 equations.

  • Plug (x,y) = (1,3) into y = ax^2+bx+c, then simplify. You should get the equation a+b+c = 3
  • Repeat for (x,y) = (2,6) and you should get 4a+2b+c = 6
  • Repeat for (x,y) = (3,11) and you should get 9a+3b+c = 11

-----------

We have this system of equations

\begin{cases}a+b+c = 3\\ 4a+2b+c = 6\\9a+3b+c = 11\end{cases}

There are a number of methods to solve this system. Substitution is what I'll go for.

Solve the first equation for c

a+b+c = 3

c = 3-a-b

Then use substitution.

4a+2b+c = 6

4a+2b+(3-a-b) = 6

3a+b+3 = 6

3a+b = 6-3

3a+b = 3

and

9a+3b+c = 11

9a+3b+(3-a-b) = 11

8a+2b + 3 = 11

8a+2b = 11-3

8a+2b = 8

We now have this reduced system of equations.

\begin{cases}3a+b = 3\\8a+2b = 8\end{cases}

I'll skip the steps as this solution is getting very lengthy as it is. The basic idea is to use substitution again. You should find that a = 1 and b = 0 form the solution set here.

Use those values to find c

c = 3-a-b

c = 3-1-0

c = 2

-----------

To summarize the previous section, we have these solutions:

a = 1, b = 0, c = 2

Therefore the equation y = ax^2+bx+c becomes y = 1x^2+0x+2 aka y = x^2+2. This lets us find any term.

Let's test it out.

  • If x = 1, then y = x^2+2 = 1^2+2 = 3
  • If x = 2, then y = x^2+2 = 2^2+2 = 6
  • If x = 3, then y = x^2+2 = 3^2+2 = 11

And so on. I'll let you test the other x values (4 through 6).

Another way to confirm the answer is to subtract 2 from each item in the original set {3,6,11,18,27,38} and you'll end up with {1,4,9,16,25,36}. This is the list of perfect squares. It shows that term x is simply x^2 but add on 2 so things are adjusted accordingly.

Side note: you can use a tool like GeoGebra or WolframAlpha to quickly solve the system of equations. However, I recommend it only as a means to check your answer rather than do the work for you.

7 0
2 years ago
The cost is $6.75 for 1 item. Find the cost of 8 items
scoray [572]

Answer:

$54

Step-by-step explanation:

Multiply the cost by the number of items

6.75x8=54

The cost is $54

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3 years ago
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Sever21 [200]
(4+2)/(1+1)= 6/2= 3 is the slope

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3 years ago
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avanturin [10]
U times 2 and seven and divide that by 3 and then dive 3 by 2 and that should leave u with three numbers.
514 and 1.5 times those together and there u go
105
6 0
2 years ago
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