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oee [108]
3 years ago
10

Give the first ten terms of the following sequences. You can assume that the sequences start with an index of 1. Logs are to bas

e 2.
(a) The n term is [log n]
(b) The n term is 2^(log n)
Mathematics
1 answer:
Vitek1552 [10]3 years ago
3 0

Answer:

a)

a1 = log(1) = 0 (2⁰ = 1)

a2 = log(2) = 1 (2¹ = 2)

a3 = log(3) = ln(3)/ln(2) = 1.098/0.693 = 1.5849

a4 = log(4) = 2 (2² = 4)

a5 = log(5) = ln(5)/ln(2) = 1.610/0.693 = 2.322

a6 = log(6) = log(3*2) = log(3)+log(2) = 1.5849+1 = 2.5849 (here I use the property log(a*b) = log(a)+log(b)

a7 = log(7) = ln(7)/ln(2) = 1.9459/0.6932 = 2.807

a8 = log(8) = 3 (2³ = 8)

a9 = log(9) = log(3²) = 2*log(3) = 2*1.5849 = 3.1699 (I use the property log(a^k) = k*log(a) )

a10 = log(10) = log(2*5) = log(2)+log(5) = 1+ 2.322= 3.322

b) I can take the results of log n we previously computed above to calculate 2^log(n), however the idea of this exercise is to learn about the definition of log_2:

log(x) is the number L such that 2^L = x. Therefore 2^log(n) = n if we take the log in base 2. This means that

a1 = 1

a2 = 2

a3 = 3

a4 = 4

a5 = 5

a6 = 6

a7 = 7

a8 = 8

a9 = 9

a10 = 10

I hope this works for you!!

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Use the factors of the function and the y-intercept to find the standard form of the equation representing Jared’s function. Typ
Georgia [21]

The y-intercept of a function is the point where the graph crosses the \mathbf{y = x^3 + 2x^2 - 193x - 270 }

  • The factors of the Jared's graph are: (x - 10), (x + 3) and (x + 9)
  • The y-intercept is -13.5
  • The standard equation of the function is: \mathbf{y = x^3 + 2x^2 - 193x - 270 }

<u>(a) The factors</u>

First, we write out the points where the function cross the x-axis.

The points are:

\mathbf{x = 10}

\mathbf{x = -3}

\mathbf{x = -9}

Equate the above points to 0

\mathbf{x -10 = 0}

\mathbf{x +3 = 0}

\mathbf{x +9 = 0}

Hence, the factors are: (x - 10), (x + 3) and (x + 9)

<u>(b) The y-intercept</u>

This is the point where the graph crosses the y-axis.

From the attached graph, the graph crosses the y-axis at -13.5.

Hence, the y-intercept is -13.5

<u>(c) The standard form</u>

In (a), we have:

\mathbf{x -10 = 0}

\mathbf{x +3 = 0}

\mathbf{x +9 = 0}

Multiply the above equations

\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 \times 0 \times 0}

\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 }

Expand

\mathbf{(x - 10) \times (x^2 + 3x + 9x + 27) = 0 }

\mathbf{(x - 10) \times (x^2 + 12x + 27) = 0 }

Expand

\mathbf{x^3 + 12x^2 + 27x - 10x^2 - 220x - 270 = 0 }

Collect like terms

\mathbf{x^3 + 12x^2 - 10x^2+ 27x  - 220x - 270 = 0 }

\mathbf{x^3 + 2x^2 - 193x - 270 = 0 }

Replace 0 with y

\mathbf{y = x^3 + 2x^2 - 193x - 270 }

Hence, the standard form is: \mathbf{y = x^3 + 2x^2 - 193x - 270 }

Read more about graphs and functions at:

brainly.com/question/18806107

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