Find the sixth term in the following arithmetic sequence.

1/4 , 1/8 ,0...

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Find the sixth term in the following arithmetic sequence.

1/4 , 1/8 ,0...

32. The ordered pairs (0, -3), (1,2), (2,7), (3,12) represent a function. What is a rule that represents this function?

The ordered ordered pairs represent a linear function (See attached graph).

The equation of the line for this case is the following:

y = 5x - 3

The rule that represents this function is linear. The line has a slope of m = 5 and has a cutoff point with the y axis which is -3. The cut point with the x axis is (3/5).

33.The size of a bee's nest increases as time passes. Your friend says that time is the dependent variable because size depends on time. Is your friend correct? Why or why not?

A dependent variable of a function f is one whose values depend on another variable, which is called an independent variable (x). It is represented by the letter y, although it is sometimes denoted like f (x).

The friend is NOT correct. In this case the dependent variable is the size of the bee's nest. The independent variable is time. This is because the size of the bee's nest increases like time passes (the size of the nest is a function of time). It can be written like

h (t) where

h: nest size (dependent variable)

t: time (independent variable).

34. A helicopter hovers 40 feet above the ground. Then the helicopter climbs at a rate of 21 feet / second. Write a rule that represents the helicopter's height, h, above the ground as a function of time, t. What is the helicopter's height after 45 seconds?

The helicopter's initial height is 40. This is unchanging, it will be represented by a constant in the height equation.

Since the helicopter climbs 21 feet per second, this can be displayed like an additional 21t. When t = 1, an additional21 feet is being added to the helicopter's height. When t = 2, an additional 42 feet is being added.

Therefore:

h = 40 + 21t

After45 seconds:

h = 40 + 21 (45)

h = 40 + 945

h = 985 feet

h = 985 feet

35. Is the following relation a function? (4.2), (1.1), (0.0), (1, -1), (4, -2). Give the domain and range.

For this case the ordered ordered pairs describe the following function:

y = (+/-) Root (x)

The domain of this function is given for all values for which x is denoted. In this case the domain is:

Domain: x> 0

The range of the function is given for all the values that "y" can take for each x. In this case the range is:

Range: y that belongs to all reals.

Note: see graphic attached.

37. Is the sequence arithmetic? 15, 14.5, 14, 13.5, 13, ... If so, give the common difference.

Arithmetic sequence is one in which the difference between each term and the next is constant.

This constant difference between successive terms is called common difference.

Each term after the first can be found by adding (or subtracting) the common difference. The formula for the nth term of an arithmetic sequence is:

an = a1 + (n-1) d

where an is the nth term, a1 is the first term and d is the common difference.

Clearing d:

d = (an - a1) / (n-1)

Substituting:

d = (13 - 15) / (5-1) = - 2/4 = -1 / 2 = -0.5

answer

d = -0.5.

38. Find the sixth term of the sequence A (n) = 6 + (n - 1) (- 2).

The sixth term of this sequence will be given by substituting the value of n = 6 in the sequence shown. Thus, the sixth term of this sequence is:

A (6) = 6 + (6 - 1) (- 2).

A (6) = 6 + (5) (- 2).

A (6) = 6 + (-10).

A (6) = 6 -10.

A (6) = -4.

answer

-4.

39. Find the range of y = -3x + 7 for the domain {-8, -6, -3, -2, 2}.

For this case the range of the function is given by all the values of and obtained within the domain [-8,2]

Therefore, the range of the function is

Range: [1,31]

NOTE: see attached graphic.

What is the vertical line test?

In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x. If a vertical line intersects a curve on an xy-plane more than once then for one value of x the curve has more than one value of y, and so, the curve does not represent a function. If all vertical lines intersect a curve at most once then the curve represents a function.

[tex]32. the ordered pairs (0,-3), (1,2), (2,7), (3,12) represent a function. what is a rule that represe[/tex]

[tex]32. the ordered pairs (0,-3), (1,2), (2,7), (3,12) represent a function. what is a rule that represe[/tex]

[tex]32. the ordered pairs (0,-3), (1,2), (2,7), (3,12) represent a function. what is a rule that represe[/tex]

The sixth term of an arithmetic progression is 37

---> a+5d = 37

5d = 37-a

sum of the first six term is 147

---> (6/2)(2a + 5d) = 147

3(2a + 37-a) = 147

a +37 = 49

a = 12

d = 5

Let [tex]a_n[/tex] denote the [tex]n[/tex]-th term in the sequence. For an arithmetic sequence, there is a constant number added to successive terms:

[tex]a_7=a_6+d[/tex]

[tex]a_8=a_7+d=a_6+2d[/tex]

[tex]a_9=a_8+d=a_6+3d[/tex]

[tex]\cdots[/tex]

[tex]a_{12}=a_{11}+d=a_6+6d[/tex]

We know [tex]a_6=\dfrac32[/tex] and [tex]a_{12}=\dfrac52[/tex], so we get

[tex]\dfrac52=\dfrac32+6d\implies 6d=1\implies d=\dfrac16[/tex]

The Next one is C

Step-by-step explanation:

This is an arithmetic progression.

Tn = a + (n -1)* d.

a = First term

n = number of terms

d = common difference.

First term, a = 12.

Sixth term, T6 = a + (6 -1)*d = a + 5d = 42

a + 5d = 42

12 + 5d = 42

5d = 42 - 12

5d = 30

d = 30/5

d = 6.

Writing the first six terms, we keep adding six.

12, 18, 24, 30, 36, 42.

Cheers.

1. Common difference= 4 . 2.The sixth term would be -7.

Step-by-step explanation: 1. 13-9=4 and 9-5=4 and 5-1=4. 2. Sixth term: 1-4=-3 And -3-4=-7.

1. m = 2.6×10⁻¹⁰

2. K < 1/2

Step-by-step explanation:

The given parameters are;

For the GP = g, gr, gr²,.....,

AP = a, (a + d), (a + 2d), (a + 3d), (a + 4d), (a + 5d)

Therefore;

g = (a + 2d)

gr = (a + 3d)

gr² = (a + 5d)

Also a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d = -6

Therefore, a = (-6 - 6d)/4

Therefore, gr - g = (a + 3d) - (a + 2d) = d

g(1 - r) = d

g = d/(1 - r)

and gr/g = r = (a + 3d)/(a + 2d)

gr²/gr = r = (a + 5d)/(a + 3d)

Hence, (a + 5d)/(a + 3d) = (a + 3d)/(a + 2d)

Substituting the value of a from above, we have

[tex]\frac{7d -3}{3d-3} = \frac{3d -3}{d-3}[/tex]

Which gives 2d² + 6d = 0

d = -3

Therefore, 4·a = -6 + 18 = 12

a = 3

g = -3

gr = -6

∴ r = 2

Therefore the term where the difference between the two terms is 10000 is found as follows;

[tex](-3)2^m - (3 + m \times (-3)) = 10000[/tex]

[tex]3m -3\cdot 2^m - 3 = 10000[/tex]

Therefore, 3·m - 3·2^m = 10003

Solving by computation, we have m = -2.6×10⁻¹⁰.

(b) For a sequence, xₙ = (2k)ᵃ

For convergence, (2k)ᵃ ÷ (2k)[tex]^{a+1}[/tex] <1

∴ 1/(2k) < 1

Hence 1/k < 2 or K < 1/2

a₆ = 54

Step-by-step explanation:

The general term of an arithmetic sequence is: [tex]$ a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \hdots $[/tex]

where, [tex]$ a_1 $[/tex] is the first term and

[tex]$ d $[/tex] is the common difference.

The [tex]$ n^{th} $[/tex] term of the sequence is given by [tex]$ a_n = a_1 + (n - 1)d $[/tex].

Therefore, [tex]$ a_6 = a_1 + (6 - 1)d = a_1 + 5d $[/tex]

[tex]$ \implies a_6 = 14 + 5(8) = 14 + 40 $[/tex]

∴ a₆ = 54

3/8

1/4 + 1/8 divided by 2