Find a_1 for the geometric sequence with the given terms. a_3 = 54 and a_5 = 486
Answers
ANSWER
[tex]6[/tex]EXPLANATION
We want to find the first term of the sequence.
The general equation for the nth term a geometric sequence is written as:
[tex]a_n=ar^{n-1}[/tex]where a = first term; r = common ratio
Let us use this to write the equations for the third term and the fifth term.
For the third term, n = 3:
[tex]\begin{gathered} a_3=ar^2 \\ \Rightarrow54=ar^2 \end{gathered}[/tex]For the fifth term, n = 5:
[tex]\begin{gathered} a_5=ar^4 \\ \Rightarrow486=ar^4 \end{gathered}[/tex]Let us make a the subject of both formula:
[tex]\begin{gathered} 54=ar^2_{} \\ \Rightarrow a=\frac{54}{r^2} \end{gathered}[/tex]and:
[tex]\begin{gathered} 486_{}=ar^4 \\ a=\frac{486}{r^4} \end{gathered}[/tex]Now, equate both equations above and solve for r:
[tex]\begin{gathered} \frac{54}{r^2}=\frac{486}{r^4} \\ \Rightarrow\frac{r^4}{r^2}=\frac{486}{54} \\ \Rightarrow r^{4-2}=9 \\ \Rightarrow r^2=9 \\ \Rightarrow r=\sqrt[]{9} \\ r=3 \end{gathered}[/tex]Now that we have the common ratio, we can solve for a using the first equation for a:
[tex]\begin{gathered} a=\frac{54}{r^2} \\ \Rightarrow a=\frac{54}{3^2}=\frac{54}{9} \\ a=6 \end{gathered}[/tex]That is the first term.
Related Questions
if m<10=77 ,m<7=47 and m<16=139, find the missing measure of m<2=?
Answers
Note that a and b are parallel lines with a transversal line of c.
<10 is congruent to <8
so :
<8 = 77
<2 and <8 are supplementary (sum of 180 degrees)
<2 + <8 = 180
<2 + 77 = 180
<2 = 103
The answer is :
<2 = 103 deg
Question 5 of 10 Which of the segments below is a secant? B D O A. CD O B. AB O C. ÃO D. BC
Answers
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
Diagram
secant = ?
Step 02:
We must analyze the diagra,m to find the solution.
Secant ===> straight line that cuts a curve in two or more parts
Segments:
CD: FALSE
AB: FALSE
AO: FALSE
BC: TRUE
The answer is:
Segment BC is a secant
17the leading term is-6x² +The expression represents aterm ispolynomial withterms. The constantand the leading coefficient is
Answers
0. quadratic
,1. two
,2. 1/7
,3. -6x²
,4. -6
1) Examining this polynomial, we can tell that:
[tex]-6x^2+\frac{1}{7}[/tex]This expression represents a quadratic polynomial with two terms. The constant term is 1/7, the leading term is -6x², and the leading coefficient is -6
2) Note that a constant term is always a number without a variable, the leading term is the one with the highest exponent, and a coefficient is a number that accompanies the leading variable.
Hay e escalones desde el pedestal hasta la cabeza de la Estatua de la Libertad. La cantidad de escalones que hay en el Monumento a Washington es 27 menos que 6 veces la cantidad de escalones que hay en la Estatua de la Libertad. ¿Qué expresión representa la cantidad de escalones que hay en el Monumento de Washington en función de e? 27 < 6e 6(e-27) 6e-27 They 6e
Answers
Let
e -----> number of steps from the pedestal to the head of the Statue of Liberty
f ----> number of steps on the Washington Monument
we have that
f=6e-27
therefore
teh answer is
6e-27
Do you understand my explanation?escalones que hay en el Monumento a Washington
we have that
f=6e-27
Evaluate the following numerical expressions.a. 2(5+(3)(2)+4)b. 2((5+3)(2+4))c. 2(5+3(2+4))Can the parentheses in any of these expressions be removed without changing the value the expression?
Answers
So,
We're going to evaluate each expression as follows:
Let's begin with a:
[tex]\begin{gathered} 2(5+(3)(2)+4) \\ 2(5+6+4) \\ 2(15) \\ =30 \end{gathered}[/tex]Now, b:
[tex]\begin{gathered} 2((5+3)(2+4)) \\ 2((8)(6)) \\ =2(48) \\ =96 \end{gathered}[/tex]And, finally, c:
[tex]\begin{gathered} 2(5+3(2+4)) \\ 2(5+3(6)) \\ 2(5+18) \\ 2(23) \\ =46 \end{gathered}[/tex]Notice that if the parentheses change, the results wouldn't be the same.
simplifying with like terms; a + 2a -7
Answers
In the expression like terms are a and 2a.
Simplify the expression
[tex]a+2a-7=3a-7[/tex]So answer is 3a - 7.
A box can be formed by cutting a square out of each corner of a piece of cardboard and folding the sides up. If the piece of cardboard is 78 cm by 78 cm and each side of the square that is cut out has length x cm, the function that gives the volume of the box is V=6084x−312x2+4x3. Complete parts (a) and (b) below.
Answers
a) Notice that:
1)
[tex]6084x-312x^2+4x^3=4x(1521-78x+x^2)=4x(x-39)^2\text{.}[/tex]Therefore V(x)=0 at x=0 and it has a double root at x=39.
2)
[tex]\begin{gathered} V(-1)=6084(-1)-312(-1)^2+4(-1)^3, \\ V(-1)=-6084-312-4<0. \end{gathered}[/tex]Therefore, V(x)<0 when x is in the following interval:
[tex](-\infty,0).[/tex]3)
[tex]V(1)=6084(1)-312(1)^2+4(1)^3>0.[/tex]Therefore, V(x)>0 when x is in the following set:
[tex](0,39)\cup(39,\infty).[/tex]b) Since x is a length, then it must be greater than zero, also 2x must be smaller than 78, therefore the values of x that makes sense in the context are in the interval:
[tex](0,39)\text{.}[/tex]Answer:
a) Option B) The values of x that makes V>0 are in the set:
[tex](0,39)\cup(39,\infty).[/tex]b) Option A) The values of x that give squares that can be cut out to construct a box are the interval:
[tex](0,39)\text{.}[/tex](0,39).
What is the average rate of change of the equation f(x)^2+3x-5 from x=2 to x=4?Type your numerical answer below. Use the hyphen (-) to represent a negative sign if necessary.
Answers
Given:
The equation is,
[tex]f\mleft(x\mright)=x^2+3x-5,x=2\text{ to x = 4}[/tex]To find: The average rate of change
Explanation:
The average rate of the change formula is,
[tex]A\left(x\right)=\frac{f\mleft(b\mright)-f\mleft(a\mright)}{b-a}[/tex]Here, we have
[tex]\begin{gathered} a=2 \\ b=4 \end{gathered}[/tex]Substituting we get,
[tex]\begin{gathered} A\lparen x)=\frac{f\mleft(4\mright)-f\mleft(2\mright)}{4-2} \\ =\frac{\left\lbrack4^2+3\left(4\right)-5\right?-\left\lbrack2^2+3\left(2\right)-5\right?}{2} \\ =\frac{16+12-5-\left\lbrack4+6-5\right\rbrack}{2} \\ =\frac{23-5}{2} \\ =\frac{18}{2} \\ =9 \end{gathered}[/tex]Final answer:
The average rate of change of the given equation is 9.
What number should be added to both sides of the following equation to solve for g?g - 18 1\3 = 2543 1\36 2\318 1\325 2\3
Answers
Answer
[tex]18\frac{1}{3}[/tex]Explanation
Given:
[tex]g-18\frac{1}{3}=25[/tex]What to find:
The number that should be added to both sides of the following equation to solve for g.
Solution:
To solve for g 18 1/3 should be added to both sides as shown below
[tex]\begin{gathered} g-18\frac{1}{3}=25 \\ \\ Add\text{ }18\frac{1}{3}\text{ }to\text{ }both\text{ }sides \\ \\ g-18\frac{1}{3}+18\frac{1}{3}=25+18\frac{1}{3} \\ \\ g=25+18\frac{1}{3} \end{gathered}[/tex]The answer is 18 1/3
a bike path is 8 miles. Max is 375% of the way to the end. How far is max on the path?
Answers
hello
the distance or length of the path is 8 miles.
Max is 375% of the way to the end. Let's find he distance of 375% on 8 miles
[tex]\begin{gathered} 375\text{ \% of 8} \\ \frac{375}{100}=\frac{x}{8} \\ \text{cross multiply both sides} \\ 100\times x=375\times8 \\ 100x=3000 \\ \frac{100x}{100}=\frac{3000}{100} \\ x=30 \end{gathered}[/tex]from the calculation above, Max is 30 miles away from the path
At a food drive, afood bank has a goal to collect 24,000 cans.If the food bank collects 100 fewer cansthan its goal, how many cans did it collect?
Answers
Given:
• Expected number of cans to collect = 24,000 cans.
,• The food bank collects 100 fewer cans.
Let's find the number of cans it collected.
Since the food bank collected 100 fewer cans, to find the number of cans it collected, let's subtract 100 from the expected goal which is 24000 cans.
Hence, we have:
Number of cans collected = expected goal - 100
Number of cans collected = 24000 - 100 = 23900
Therefore, the number of cans the food bank collected was 23,900 cans.
ANSWER:
23900 cans
Using the drawing find y: MzBDJ = 7y + 2, mZJDR = 2y + 7 * HRT O 25 O 19 O O TY
Answers
Leo is 5 years older than Pat. In 10 years leo will be twice pat's present age. How old is leo
Answers
Answer:
20
Step-by-step explanation:
Pat is 15.
Leo is 5 years older(20)
Leo will be 30 in 10 years.
15 x 2 = 30
given triangle CAT is congruent to triangle DOG. Solve for x
Answers
Answer:
The value of x is 2 or -5.5.
Explanation:
Given that triangle CAT is congruent to triangle DOG, then we have that:
[tex]\angle T\cong\angle G[/tex]Step 1: We determine the value of angle T.
[tex]\begin{gathered} \angle T=180^0-(87^0+75^0) \\ =180^0-162^0 \\ =18^0 \end{gathered}[/tex]Step 2: Since angle T is congruent to angle G, then:
[tex](x+4)(2x-1)=18[/tex]Step 3: We solve the equation above for x.
[tex]\begin{gathered} 2x^2-x+8x-4-18=0 \\ 2x^2+7x-22=0 \\ 2x^2-4x+11x-22=0 \\ 2x(x-2)+11(x-2)=0 \\ (2x+11)(x-2)=0 \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} 2x+11=0\text{ or x-2=0} \\ 2x=-11\text{ or x=2} \\ x=-5.5\text{ or x=2} \end{gathered}[/tex]The value of x is 2 or -5.5.
For each coefficient choose whether it is positive or negative. Choose the coefficient with the greatest value. Choose the coefficient closest to zero.
Answers
First we need to find if the coefficients are negative or positive. The function:
[tex]\lvert x\rvert[/tex]Is always positive which means that its graph must be over the x axis. If this function is multiplied by a positive coefficient then the graph remains over the x axis. On the other hand, if it's multiplied by a negative number then the graph is now under the x axis. A and B graph are over the x axis so they are positive whereas C and D graphs are under the x axis and they are negative and that's the answer for a.
Then we must find the coefficient with the greatest value. Since a positive number is greater than any negative number we can discard C and D. Now we have two options, A and B which we know are different numbers since their graph are different. Both are V shaped but graph B is sharper than graph C. This means that B is greater than C. Then, the answer to part b is coefficient B.
In part c we must choose the coefficient that is closest to 0. Using the same argument as before, the sharper the V shaped graph is the greatest absolute value its coefficient has. This means that the least sharp graph is that of the coefficient that is closer to 0. Looking at the four graphs you can see that the least sharp V is that of coefficient A. Then, the answer to part c is coefficient A.
1 point 3 John ran 3 les in of an hour Marlon ran s 4 miles in of an hour How lar did Marlon run in one hour?
Answers
Answer:
6.2 miles per hour
Explanation:
Marlon ran 8 1/4 miles in 4/3 of an hour.
So, we first need to transform the mixed number 8 1/4 into a fraction using the following equation:
[tex]\begin{gathered} A\frac{b}{c}=\frac{A\cdot c+b}{c} \\ 8\frac{1}{4}=\frac{8\cdot4+1}{4}=\frac{32+1}{4}=\frac{33}{4} \end{gathered}[/tex]Then, we need to divide 33/4 miles by 4/3 hour as:
[tex]\begin{gathered} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot d}{b\cdot c} \\ \frac{\frac{33}{4}}{\frac{4}{3}}=\frac{33\cdot3}{4\cdot4}=\frac{99}{16}=6.2\text{ miles per hour} \end{gathered}[/tex]So, Marlon ran 6.2 miles per hour.
If w = 18, what is the value of 3w − 11?
(A) 307
(B) 65
(C) 43
(D) 10
Answers
Answer: 43 ==>
Step-by-step explanation: 3w − 11=3(18)-11=54-11=43 ==> C.
Given that,
w = 18
3w - 11 = ?
3(18) - 11
54 - 11
43
correct option is (c) 43
Joey buys a home for $205,900. His home is predicted to increase in value 4% each year. What is the predicted value of his home in 22 years? Round answer to thenearest whole number
Answers
Since every year the value of the house increase by 4%, the new value will be the previous value plus 4% of the previous value. To find 4% of a quantity, we just have have to multiply it by 4 and then divide by 100(or, written as a decimal, multiply the number by 0.04).
If we call the previous value of the house as P and the new value as N, the new value after one year will be
[tex]N=P+0.04P=(1+0.04)P=1.04P[/tex]Every year that passes, to get the new value we multiply again by 1.04. The expression for the predicted value after t years is
[tex]N(t)=P_0(1.04)^t[/tex]Where P0 represents the initial value of the house. Evaluating t = 22 on this expression, we have
[tex]N(22)=205,900(1.04)^{22}=487,966.279171\ldots\approx487,966[/tex]The predicted value of his home in 22 years is $487,966.
A figure is made up of two triangles and a square. The trianglesand the square have the same base length of 9 feet. Thetriangles have a height of 12.3 feet. What is the total area of thefigure?
Answers
I am having trouble trying to figure out letter B
Answers
The confidence level is [0.1883, 0.3866] and the 95% confidence interval for the cost is [0.25x 564.9, 0.25x1160.1] = [141.23, 290]
n = 80
a) p = 23/80 = 0.2875
% = 95
Standard error SE = √(p(1 - p)/n = √(0.2875(1 - 0.2875)/80)
z - score = 1.9599
Width of the confidence level = z x SE = 1.95996 x 0.05060 = 0.0991
Lower level of the confidence level = p - width = 0.2875 - 0.099177 = 0.1883
upper level of the confidence level = p + width = 0.2875 + 0.099177 = 0.3866778
The confidence level is [0.1883, 0.3866]
b) The 95% confidence level for the number of such customers = [0.1883 x 3000, 0.3866 x 3000] = [564.9, 1160.1]
The 95% confidence interval for the cost = [0.25x 564.9, 0.25x1160.1] = [141.23, 290]
Therefore, the confidence level is [0.1883, 0.3866] and the 95% confidence interval for the cost is [0.25x 564.9, 0.25x1160.1] = [141.23, 290]
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What is the domain of the rational function f of x is equal to the quantity x squared plus x minus 6 end quantity over the quantity x cubed minus 3 times x squared minus 16 times x plus 48 end quantity question mark
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To solve this problem, we have to find the zeros of the expression in the denominator, to do it, factor the expression:
[tex]\begin{gathered} x^3-3x^2-26x+48=0 \\ (x-3)(x-4)(x+4)=0 \\ x-3=0 \\ x=3 \\ x-4=0 \\ x=4 \\ x+4=0 \\ x=-4 \end{gathered}[/tex]The zeros of the function are x=3,4,-4.
Since these values make the expression be zero, they are not included in the domain of the function. This is because the expression in the denominator can not be zero, otherwise, the function would be undefined.
The correct answer is B:
[tex]\mleft\lbrace x\in R\mright|x\ne-4,3,4\}[/tex]I'll just send you the picture. there's too much to type
Answers
ANSWER
[tex]\text{ \$278.75}[/tex]EXPLANATION
We have that Sammi has $125.75 in her account and deposits (adds) $25.50 every month for 6 months.
To find how much is there after 6 months, first, find out how much she added to the account and then add that to the initial amount that was there.
After 6 months she deposited:
[tex]\begin{gathered} 6\cdot25.50 \\ \text{ \$153} \end{gathered}[/tex]Now, add that to the initial amount there:
[tex]\begin{gathered} 125.75+153 \\ \text{ \$278.75} \end{gathered}[/tex]That is the amount in the account at the end of 6 months.
is 1,000 feet greater than 300 yards
Answers
hello
to solve this question, we have to know the the dimensions or
What are the coordinates of the y- intercept of this function y 6 -5 1 2
Answers
The coordinates of the y-intercept of the function can be obtained if we follow the steps below
Step 1: we can get the equation of the graph using the equation below
[tex]\frac{y_2-y_1}{x_2\text{ -}x_1}\text{ = }\frac{y\text{ -}y_1}{x\text{ -}x_1}[/tex]Step 2: Select two coordinates that will be substituted into the equation
Selecting the points
(-2, -6) and (-1, -5)
[tex]\frac{-5\text{ -(-6)}}{-1\text{ -(-2)}}\text{ =}\frac{y-(-6)}{x\text{ -(-2)}}[/tex][tex]\frac{1}{1}=\frac{y+6}{x+2}[/tex]Cross multiplying
y + 6 = x + 2
y = x + 2 - 6
y = x -4
The equation of the line is y = x - 4
The next step is to obtain the y-intercept which is obtained by substituting x = 0 into the equation of the line.
If x = 0
then y = 0 - 4
Hence y = -4
Therefore the coordinates of the y-intercept are (0, -4)
I need help with this question. Including the breakdown. Thank you
Answers
Let
x -------> the height of the tower
y -----> the height of the guy wire
we have that
x=y-3 ------> equation 1
y=9 m
substitute the value of y in equation 1
x=9-3
x=6 m
therefore
the height of the tower is 6 metersA plant is already 42 centimeters tall, and it will grow one centimeter every month.Let H be the plant's height (in centimeters) after M months.Write an equation relating H to M. Then use this equation to find the plant's helght after 35 months.Equation:D-ORХ5?Plant's height after 35 months: centimeters
Answers
write the height of the plant after m months as a linear function in which the y-intercept is the plant's initial height (42 cm) and the slope is the growth the plant experiences after each month (1 cm)
[tex]H=M+42[/tex]then, after 35 months
[tex]\begin{gathered} H=35+42 \\ H=77 \end{gathered}[/tex]The plant's height after 35 months is 77 cm.
Shaan and anita are married and have two children, ages 3 and 9. anita is a "nonworking" spouse who devotes all of her time to household activities. estimate how much life insurance shaan and anita should carry.
Answers
The cost of life insurance that shaan and anita should carry would be = $150,000
What is life insurance policy?A life insurance policy is defined as the type of insurance an individual contracts so as to enable their beneficiaries a stipulated amount of money after the individual dies.
The number of children owned by shaan and anita = 2
The age of the youngest child = 3
The age of the eldest child = 9
The insurance policy need = Number of years until the youngest child is 18 × $10,000
The number of years until the youngest child is 18 = 18-3= 15
15× 10,000 = $150,000
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A quadratic equation is shown below:x^2 + 18x + 76 = 0Which of the following is the first correct step to write the above equation in the form (x-p)^2 = q, when p and q are integers? A. Add 9 to both sides of the equationB. Add 5 to both sides of the equationC. Subtract 5 from both sides of the equationD. Subtract 9 from both sides of the equation.
Answers
SOLUTION:
[tex]\begin{gathered} x^2\text{ + 18x + 76 = 0} \\ To\text{ make left hand p}\operatorname{erf}ect\text{ square, we add 5 to both sides} \\ x^2\text{ + 18x + 76 + 5 = 0 + 5} \\ x^2\text{ + 18x + 81 = 5} \\ x^2\text{ }+18x+9^2\text{ = 5} \\ (x+9)^2\text{ = 5} \end{gathered}[/tex]The correct option B, that is, add 5 to both sides of the equation.
can you please help me before I get on error message and get kicked out
Answers
We have the following:
What we must do is calculate the total rate, that is, add all of them and then calculate each part, that is:
[tex]4+5+5+8+9+9=40[/tex]Now we calculate each angle like this
[tex]\begin{gathered} 720\cdot\frac{4}{40}=72 \\ 720\cdot\frac{5}{40}=90 \\ 720\cdot\frac{8}{40}=144 \\ 720\cdot\frac{9}{40}=162 \\ we\text{ add} \\ 72+90+90+144+162+162=720 \end{gathered}[/tex]then we can affirm that the smallest angle is 72 °
I have a pentagon that 8 and 8 and 6 and 6 and 10 whats the perimeter?
Answers
The perimeter of any closed figure is equal to sum of all sides of the figure.
Determine the perimeter of pentagon by addition of measure of sides of pentagon.
[tex]\begin{gathered} P=8+8+6+10+6 \\ =38 \end{gathered}[/tex]So answer is 38 cm.