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3 years ago

Find the 98th term for the arithmetic sequence 3,4.2,5.4,6.6,7.8,...

Mathematics

1 answer:

prisoha [69]3 years ago

6 0

A₁ = 3
a₂ = 4.2

d = a₂ - a₁
= 4.2 - 3
= 1.2

a₉₈ = a₁ + d(n-1)
= 3 + 1.2(98-1)
= 3 + 1.2×97
= 3 + 116.4
= 119.4

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Sally is having a party and she has invited nine of her friends.

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Answer:

1) 5 limes, 5 lemons, 1,000 mL of pineapple juice, 2,000 mL of lemonade, 1,000 of orange juice

2) Limes = $2, Lemons = $2.5, Lemonade = $1.60, Orange Juice = 80c, Pineapple Juice = 90c

Step-by-step explanation:

1/2 limes · 10 = 5

1/2 lemons · 10 = 5

100 mL of pineapple juice · 10 = 1,000 mL

200 mL of lemonade · 10 = 2,000 mL

100 mL of orange juice · 10 = 1,000 mL

Cost:

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5 lemons · 0.5 = $2.5

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1,000 mL of pineapple juice = 1 L, this costs 90¢

2,000 mL of lemonade = 2 L, this costs $1.60

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Overall:

5 limes for $2

5 lemons for $2.5

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2 L of lemonade for $1.60

1 L of orange juice for 80¢

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4 years ago

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3 years ago

Determine whether the set of all linear combinations of the following set of vector in R^3 is a line or a plane or all of R^3.a.

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Answer:

a. Line

b. Plane

c. All of R^3

Step-by-step explanation:

In order to answer this question, we need to study the linear independence between the vectors :

1 - A set of three linearly independent vectors in R^3 generates R^3.

2 - A set of two linearly independent vectors in R^3 generates a plane.

3 - A set of one vector in R^3 generates a line.

The next step to answer this question is to analyze the independence between the vectors of each set. We can do this by putting the vectors into the row of a R^(3x3) matrix. Then, by working out with the matrix we will find how many linearly independent vectors the set has :

a. Let's put the vectors into the rows of a matrix :

$\left[\begin{array}{ccc}-2&5&-3\\6&-15&9\\-10&25&-15\end{array}\right]$ ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒

$\left[\begin{array}{ccc}-2&5&-3\\0&0&0\\0&0&0\end{array}\right]$

We find that the second vector is a linear combination from the first and the third one (in fact, the second vector is the first vector multiply by -3).

We also find that the third vector is a linear combination from the first and the second one (in fact, the third vector is the first vector multiply by 5).

At the end, we only have one vector in R^3 ⇒ The set of all linear combinations of the set a. is a line in R^3.

b. Again, let's put the vectors into the rows of a matrix :

$\left[\begin{array}{ccc}1&2&0\\1&1&1\\4&5&3\end{array}\right]$ ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒

$\left[\begin{array}{ccc}1&1&1\\0&1&-1\\0&0&0\end{array}\right]$

We find that there are only two linearly independent vectors in the set so the set of all linear combinations of the set b. is a plane (in fact, the third vector is equivalent to the first vector plus three times the second vector).

c. Finally :

$\left[\begin{array}{ccc}0&0&3\\0&1&2\\1&1&0\end{array}\right]$ ⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒

$\left[\begin{array}{ccc}1&1&0\\0&1&2\\0&0&3\end{array}\right]$

The set is linearly independent so the set of all linear combination of the set c. is all of R^3.

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3 years ago