Proofs with transformations

Describe and correct the error in finding the tenth term of the arithmetic sequence 4, 12, 20, 28.

ycow [4]

Answer: The tenth term is 76

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Explanation:

We use this arithmetic sequence formula to get the nth term

$a_n = a_1 + d(n-1)$

Plug in $a_1 = 4, \ d = 8, \ n = 10$ and you should get the following:

$a_n = a_1 + d(n-1)\\\\a_{10} = 4 + 8(10-1)\\\\a_{10} = 4 + 8(9)\\\\a_{10} = \textbf{76}\\\\$

The tenth term is <u>76</u>

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We can verify this by listing out the terms one by one. Start at 4, add on 8 each time, until you generate the 10th term. A table like the one shown below is a good way to keep track of all the terms.

$\begin{array}{c|c}\boldsymbol{n} & \boldsymbol{a_n}\\1 & 4\\2 & 12\\3 & 20\\4 & 28\\5 & 36\\6 & 44\\7 & 52\\8 & 60\\9 & 68\\10 & \textbf{76}\\\end{array}$

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In short, the error is with the "10" in the expression 4+10(8). The student should have used 9 instead. This is because of the n-1 term in $a_n = a_1 + d(n-1)$ which shifts everything one spot to the left.

7 0

2 years ago

Kami is cooking dinner for a family reunion that 21 people are attending. If her recipe calls for 1.6 ounces of ground beef per

givi [52]

Answer:

Kami will need 33.6 ounces of ground beef for the recipe

Step-by-step explanation:

From the problem, we know that one person will use up 1.6 ounces of ground beef. We also know that there are 21 people that will be attending the party. Each of those 21 people will definitely use up 1.6 ounces of ground beef.

Hence, for the recipe to be perfect, Kami will have to use 21 X 1.6 ounces = 33.6 ounces of ground to make the recipe for the entire dinner.

3 0

3 years ago

The functions f and g are defined by f: x = 4 - x and g: x = hx² + k. If the composite function gf is given by gf: x + 2x² - 16x

kozerog [31]

Answer:

Step-by-step explanation:

f(x) = 4-x

g(x) = h $x^{2}$ +k

g(f(x)) = 2 $x^{2}$ -16x+26

so put f(x) in g(x)

h $(4-x)^{2}$ +k

h((4-x)(4-x) + k

h( $x^{2}$ -8x+16)+k

if h = 2 , then

2 $x^{2}$ -16x+32 + k

and we want 26 instead of 32 so subtract 6 so K = (-6)

2 $x^{2}$ -16x+32 + (-6)

2 $x^{2}$ -16x+32 - 6

2 $x^{2}$ -16x+26

h=2

k=(-6)

5 0

2 years ago

No real numbers are irrational numbers

qaws [65]

True. A rational number is a number that can be put into a fraction and all real numbers can in the form of x/1

7 0

3 years ago

Read 2 more answers

Observe that equation (3) has constant coefficients. If y1(x) and y2(x) form a fun- damental set of solutions of equation (3), s

WARRIOR [948]

The equations (2) and (3) you referred to are unavailable, but it is clear that you are trying to show that two set of solutions y1 and y2, to a (second-order) differential equation are solutions, and form a fundamental set. This will be explained.

Answer:

SOLUTION OF A DIFFERENTIAL EQUATION.

Two functions y1 and y2 are set to be solutions to a differential equation if they both satisfy the said differential equation.

Suppose we have a differential equation

y'' + py' + qy = r

If y1 satisfies this differential equation, then

y1'' + py1' + qy1 = r

FUNDAMENTAL SET OF DIFFERENTIAL EQUATION.

Two functions y1 and y2 are said to form a fundamental set of solutions to a second-order differential equation if they are linearly independent. The functions are linearly independent if their Wronskian is different from zero.

If W(y1, y2) ≠ 0

Then solutions y1 and y2 form a fundamental set of the given differential equation.

7 0

3 years ago