if you are given a3=2 a5=16, find a100.
Answers
I suppose \(a_n\) denotes the n-th term of some sequence, and we're given the 3rd and 5th terms \(a_3=2\) and \(a_5=16\). On this information alone, it's impossible to determine the 100th term \(a_{100}\) because there are infinitely many sequences where 2 and 16 are the 3rd and 5th terms.
To get around that, I'll offer two plausible solutions based on different assumptions. So bear in mind that this is not a complete answer, and indeed may not even be applicable.
• Assumption 1: the sequence is arithmetic (a.k.a. linear)
In this case, consecutive terms differ by a constant d, or
\(a_n = a_{n-1} + d\)
By this relation,
\(a_{n-1} = a_{n-2} + d\)
and by substitution,
\(a_n = (a_{n-2} + d) + d = a_{n-2} + 2d\)
We can continue in this fashion to get
\(a_n = a_{n-3} + 3d\)
\(a_n = a_{n-4} + 4d\)
and so on, down to writing the n-th term in terms of the first as
\(a_n = a_1 + (n-1)d\)
Now, with the given known values, we have
\(a_3 = a_1 + 2d = 2\)
\(a_5 = a_1 + 4d = 16\)
Eliminate \(a_1\) to solve for d :
\((a_1 + 4d) - (a_1 + 2d) = 16 - 2 \implies 2d = 14 \implies d = 7\)
Find the first term \(a_1\) :
\(a_1 + 2\times7 = 2 \implies a_1 = 2 - 14 = -12\)
Then the 100th term in the sequence is
\(a_{100} = a_1 + 99d = -12 + 99\times7 = \boxed{681}\)
• Assumption 2: the sequence is geometric
In this case, the ratio of consecutive terms is a constant r such that
\(a_n = r a_{n-1}\)
We can solve for \(a_n\) in terms of \(a_1\) like we did in the arithmetic case.
\(a_{n-1} = ra_{n-2} \implies a_n = r\left(ra_{n-2}\right) = r^2 a_{n-2}\)
and so on down to
\(a_n = r^{n-1} a_1\)
Now,
\(a_3 = r^2 a_1 = 2\)
\(a_5 = r^4 a_1 = 16\)
Eliminate \(a_1\) and solve for r by dividing
\(\dfrac{a_5}{a_3} = \dfrac{r^4a_1}{r^2a_1} = \dfrac{16}2 \implies r^2 = 8 \implies r = 2\sqrt2\)
Solve for \(a_1\) :
\(r^2 a_1 = 8a_1 = 2 \implies a_1 = \dfrac14\)
Then the 100th term is
\(a_{100} = \dfrac{(2\sqrt2)^{99}}4 = \boxed{\dfrac{\sqrt{8^{99}}}4}\)
The arithmetic case seems more likely since the final answer is a simple integer, but that's just my opinion...
Related Questions
Rita is hiking along a trail that is 14.8 miles long. So far she has hiked along one-tenth of the trail.
How far has Rita hiked?
Answers
Answer:
1.48 miles
Step-by-step explanation:
If the trail is 14.8 miles and Rita has hiked 1/10 of it. Then just multiply 14.8 by 1/10.
14.8 * 1/10 = 1.48 miles.
Answer:
1.48
Step-by-step explanation:
I know the answer is correct
What is 48% of 48?
(Express your answer rounded correctly to one decimal place.)
Answers
Answer:
it is a ratio from a hole number
Step-by-step explanation:
get 48 % as a nuber by dividing it by 100
then multiply the answer by 48 and you are done
48% in decimal is
\(\dfrac{48}{100}= 0.48\)
Now we multiply 0.48 by 48 and so:
\(48\times0.48=23.04\)
Thus, 48% of 48 is 23.04
Pre college need withh
Answers
Answer:
Step-by-step explanation:
f(9) = |9 + 9| = |18| = 18
f(-4) = |-4 + 9| = |5| = 5
f(0) = |0 + 9| = |9| = 9
pretty boring examples as you're always taking the absolute value of a positive number.
what do you suppose f(-12) is???
f(-12) = |-12 + 9| = |-3| = 3
Kiara invested $3,500 into two accounts. One account paid 5% interest and the other paid 7.5% interest. She earned 6%
interest on the total investment. How much money did she put in each account?
Answers
Answer:
A=$2100
B=$1400
Step-by-step explanation:
3500*1.06=3710
a+b=3500
1.05a+1.075b=3710
1.05a+1.05b=3675
1.05a+1.075b=3710
Subtract 1.05a+1.05b=3675 from 1.05a+1.075b=3710
0.025b=35
b=1400
a=3500-1400=2100
Double check: (2100*1.05)+(1400*1.075)=3710
3500*1.06=3710
The amount invested in the account that pays a 5% interest is $2100 and the amount invested in the account that pays a 7.5% interest is $1400.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
0.05a + 0.075b = (0.06 x $3500)
0.05a + 0.075b = $210 ----- equation 1
a + b = $3,500 -------- equation 2
Where:
a is the amount invested in the account that paid a 5% interest.
b is the amount invested in the account that paid a 7.5% interest.
The amount invested in the account that pays a 7.5% interest.
In order to determine this value, multiply equation 2 by 0.05
0.05a + 0.05b = 175 ------- equation 3
0.025b = 35
b = $1400
The amount invested in the account that pays a 5% interest.
a + $1400 = $3500
a = $3500 - 1400
a = $2100
Therefore, The amount invested in the account that pays a 5% interest is $2100 and the amount invested in the account that pays a 7.5% interest is $1400.
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Im an older woman and i could really use some help this math is new to meTwo angles are supplementary. One angle is 36° more than the other. Find the measures of the angles.
Answers
To solve this, we need to create a system of equations using the info provided by the problem.
Two supplementary angles are two angles that add up to 180º
If we call one angle x and the other y:
\(x+y=180º\)Next, one angle is 36º more than the other, we can write then:
\(x+36º=y\)Now we have a system of equations. We can then use substitution to get the values of each angle. Substitution consist in substitute the value of one equation in to the other, since we know, by the second equation, that y = x + 36º, we can replace this value of y in the first eqaution:
\(\begin{gathered} \begin{cases}x+y=180º \\ y=x+36º\end{cases} \\ x+x+36º=180º \end{gathered}\)Then we can solve for x:
\(\begin{gathered} x+x+36º=180º \\ 2x=180º-36º \\ x=\frac{144º}{2} \\ x=72º \end{gathered}\)The measure of one of the angles is 72º. To find the measure of the other angle, we can go back to the first equation and substitute the value of x = 72º:
\(\begin{gathered} \begin{cases}x+y=180º \\ x=72º\end{cases} \\ 72º+y=180º \end{gathered}\)And solve for y:
\(\begin{gathered} 72º+y=180º \\ y=180º-72º=108º \end{gathered}\)THe value of the other angle is 108º
Now we can verify if the values we gat are correct by using the second equation. Angle x plus 36º must to be equal to angle y:
\(72º+36º=108º\)Thus, the answer we get is correct.
Explain why the simultaneous equations y = -2/3x + 4/3 and 12x + 18y = 24 have an infinite number of solutions. What can you deduce/conclude about the two straight lines representing the equations?
Answers
Answer:
It's the same equation
Step-by-step explanation:
12x+18y=24
y=(-12x+24)/18
y=-2/3x+4/3
The intersect is the solution, so there are infinitely many solutions
Never mind i got it!
Answers
Answer:
OOk
Step-by-step explanation:
Mr.West ate grapes for 11 days. He ate 2 1/11 ounces of grapes each day . How many grapes did he eat ?
Answers
Mr.West ate 23 ounces of grapes in 11 days by eating 2 1/11 ounces of grapes each day.
What is a fraction?A fraction is written in the form of p/q, where q ≠ 0.
Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.
Given, Mr.West ate grapes for 11 days. He ate 2 1/11 ounces of grapes each day.
Therefore, The total number of grapes Mr.West ate is,
= 11×(2 1/11) grapes.
= 11×(23/11) grapes.
= 23 grapes.
So, He ate 23 ounces of grapes in 11 days.
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You measure the period of a mass oscillating on a vertical spring ten times as follows:
Period (s): 1.06, 1.31, 1.28, 0.99, 1.48, 1.37, 0.98, 1.31, 1.59, 1.55
Required:
What are the mean and (sample) standard deviation?
a. Mean: 1.228, Standard Deviation: 0.2135
b. Mean: 1.325, Standard Deviation: 0.1674
c. Mean: 1.292. Standard Deviation: 0.2211
d. Mean: 1.228, Standard Deviation: 0.2098
e. Mean: 1.292, Standard Deviation: 0.2135
Answers
Answer:
Step-by-step explanation:
Mean is the ratio of sum of the dataset to the sample size. Mathematically:
\(\overline x = \frac{\sum Xi}{N}\)
Xi are the individual periods
N is the sample size
\(\sum Xi = 1.06+1.31+1.28+0.99+1.48+1.37+0.98+1.31+1.59+1.55\\\sum Xi = 12.92\)
N = 10
Substitute
\(\overline x = \frac{12.92}{10}\\\overline x = 1.292\)
hence the mean of the samples is 1.292
For the standard deviation:
\(\sigma = \sqrt{\frac{\sum (x-\overline x)^2}{N} } \\\)
\(\sum (x-\overline x)^2 = (1.06-1.292)^2+ (1.31-1.292)^2+ (1.28-1.292)^2+ (10.99-1.292)^2+ (1.48-1.292)^2+ (1.37-1.292)^2+ (0.98-1.292)^2+ (1.31-1.292)^2+ (1.59-1.292)^2+ (1.55-1.292)^2\\\sum (x-\overline x)^2 = 0.43996\)
Substitute into the formula:
\(\sigma = \sqrt{\frac{0.43996}{10} }} \\\sigma = \sqrt{{0.043996}}} \\\sigma = 0.2098\)
Hence the standard deviation is 0.2098
How many quarts rounded to the nearest tenth are in 50 liters
Answers
Answer:
52.8
Step-by-step explanation:
If you were to look this up 50 liters = 52.8344
The question asks you to round to the nearest tenth so 52.8344 is now shown as 52.83 when you round that you can see that 52.83 rounded to the nearest tenth is 52.8 and that's your answer.
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An ice cream factory makes 160 quarts of ice cream in 5 hours. How many
quarts could be made in 36 hours? What was that rate per day?
Answers
Answer:
1152
Step-by-step explanation:
5 hours = 160 quarts
1 hour = 32 quarts
and finally
36 hours = 32×36
= 1152
each function
f(x)=-4x-5;
ion for
Find ƒ(1)
for the given
Answers
When x is equal to 1, the Function f(x) = -4x - 5 yields a value of -9.
The find ƒ(1) for the function f(x) = -4x - 5, we need to substitute x = 1 into the function and evaluate the expression.
Replacing x with 1, we have:
ƒ(1) = -4(1) - 5
Simplifying further:
ƒ(1) = -4 - 5
ƒ(1) = -9
Therefore, when x is equal to 1, the value of the function f(x) = -4x - 5 is ƒ(1) = -9.
Let's break down the steps taken to arrive at the solution:
1. Start with the function f(x) = -4x - 5.
2. Replace x with 1 in the function.
3. Evaluate the expression by performing the necessary operations.
4. Simplify the expression to obtain the final result.
In this case, substituting x = 1 into the function f(x) = -4x - 5 gives us ƒ(1) = -9 as the output.
It is essential to note that the notation ƒ(1) represents the value of the function ƒ(x) when x is equal to 1. It signifies evaluating the function at a specific input value, which, in this case, is 1.
Thus, when x is equal to 1, the function f(x) = -4x - 5 yields a value of -9.
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find the inverse of each function
Answers
Answer:
c
Step-by-step explanation:
assume base 10
-logy = x
\( \frac{1}{ log(y) } = x\)
log base x y = 5 turns into the format x^ 5 = y
implement that to get c
Bella, Tia, and Ron participate in a trivia contest at a book shop. Tia scores 4 points, Ron scores 3 points, and Bella scores 2 points. Each player gets a number of paperback books equal to the square of the points he or she scores. What is the total number of books the players receive?
42 × 32 × 22
42 + 32 + 22
24 × 23 × 22
24 + 23 + 22
Answers
Answer:
4(2 on top) + 3(2 on top) + 2(2 on top)
Step-by-step explanation:
Question 13
Which of the following best describes the rule for the input-output table
below?
O A P(h) -
B. P(b)-4+h
C. P(h)-4.h
D. P(h)-3-h
h
157
9
11
13
115
17
P(h)
20
28
36
8831
44
52
60
68
G
Answers
The linear function that describes the data in the table for this problem is given as follows:
C. P(h) = h + 11.
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.From the table, we have that when h increases by 1, P(h) also increases by 1, hence the slope m is given as follows:
m = 1/1 = 1.
Hence:
P(h) = h + b.
When h = 4, P(h) = 15, hence the intercept b is given as follows:
15 = 4 + b
b = 11.
Hence the rule is:
P(h) = h + 11.
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.
Simplify.
{[2 + 3(−3 – 4) – 3] – 2} ÷ 23
Answers
Answer:
-0.2
Step-by-step explanation:
pls give me brainliest if possible
Which equation has the solutions x=1+/- 5?
Ox^2+2x + 4 =0
Ox^2-2x + 4 =0
Ox^2+ 2x-4 = 0
Ox^2-2x – 4 =0
Answers
Answer:
\(x^{2} -2x-4=0\)
Solving Steps
Use the quadratic formula
\(x=\frac{-(-2)(+/-)\sqrt{(-2)^2-4*1*(-4)} }{2*1}\)
Multiply / Remove the parentheses / Evaluate
\(x=\frac{2(+/-)\sqrt{4+16} }{2}\)
Calculate
\(x=\frac{2(+/-)\sqrt{20} }{2}\)
Separate the solutions
\(x=\frac{2(+/-)2\sqrt{5} }{2} \\x=\frac{2-2\sqrt{5} }2}\)
Simplify
\(x=1+\sqrt{5} \\x=1-\sqrt{5}\)
Yael used to have a square garage with 222 ft2 of floor space. She recently built an addition to it. The garage is still a square, but now it has 50% more floor space. What was the length of one side of the garage originally? What is the length of one side of the garage now
Answers
• The length of one side of the garage originally is approximately 14.9 ft
,• The length of one side of the garage now is approximately 18.3 ft
Explanation:Old garage = 222 sq. ft
New garage has 50% more floor space.
50% of 222 = 111
Therefore, the new garage is (222 + 111) sq. ft = 333 sq. ft
Since the garage is square,
one side of the old garage is:
\(\sqrt[]{222}=14.9\text{ ft}\)One side of the new garage is:
\(\sqrt[]{333}=18.3\text{ ft}\)5,10,15,20,25 what is the common difference
Answers
Answer:
The common difference in the given sequence is 5.
Step-by-step explanation:
The common difference in the sequence is 5 because they all add by 5 each time for example 5 + 5= 10 and 10+5=15
9. An airline food service finds that a recent flight of 120 passengers ordered dinner as depicted in the table below. Based on this data, how many lasagna dinners are likely to be needed for a flight with 380 passengers?
Dinner Frequency
Chicken 63
Fish 30
Lasagna 25
Vegetarian 2
A. 80 B. 85 C. 206 D. 301
Answers
Answer:
80
Step-by-step explanation:
25/120 ( since you are trying to find probability of a situation)=.2083
.2083*380=79.167 (just round it off to the nearest 1 which is 80).
which of the following are trinomials?
PLS, I REALLY NEED THIS ANSWER.
Answers
Answer:
Option (A) and Option (B)
Step-by-step explanation:
Trinomial is a polynomial having three terms.
Option A
\(\frac{5}{7}y^3+5y^2+y\)
Number of terms = 3
So it's a trinomial.
Option (B)
x⁴ + x² + 1
Number of terms = 3
It's a trinomial.
Option C
6x² + 0.5y²
Number of terms = 2
It's a binomial.
Option D
x² + 3
Number of terms = 2
It's a binomial.
Option E
x¹¹
Number of terms = 1
Monomial
Option F
8x
Monomial.
Therefore, Option (A) and Option (B) are the correct options.
Find all complex solutions (real and non-real). (Enter your answers as a comma-separated list.) x3 8x2 21x 20
Answers
Answer:
The answer is "non-real solution: "i-3 and -i-3" and real solution is -2".
Step-by-step explanation:
In the given equation there is some mistype error so, its correct equation can be defined as follows:
Given value:
\(\bold{ f(x) = x^3+ 8x^2 +22x +20}\)
use hit and trial method we put x value are -2:
\(f(-2) = (-2)^3+ 8(-2)^2 +22(-2) +20\\\\\)
\(= -8 + 8 \times 4 +22 \times -2 +20\\\\= -8 + 32 -44 +20\\\\= -52 + 52\\\\=0\)
So, the real solution to the equation is "-2".
In the above solution, we calculate the f(x) =-2 is its real solution. so, the calculation for non-real solution:
divide the value \(x^3+8x^2+ 22x +20\) by (\(x+2\)):
Quotient: \(x^2+6x+10\)
Remainder: 0
\(\to x^2+6x+10 \\\\compare \ with \ ax^2+bx+c: \\\\a= 1\\b= 6\\c=10\\\\\bold{Formula:}\\\\\to D= b^2-4ac\\\\\)
\(= 6^2- 4 \times 1 \times 10\\\\ = 36- 40\\\\ = -4\)
Formula for calculate non-real solution:
\(\bold{\frac{-b\pm \sqrt{D}}{2}}\)
non-real solution: "i-3 and -i-3"
The managers at Smith's BBQ have been asked to cut back Total Wages by 2.5% in the next two
weeks to account for slow sales. Using the Current Week as the benchmark, what would be the
estimate for the reduced Total Wages for the next two weeks combined?
Answers
The required estimate for the reduced Total Wages for the next two weeks combined would be approximately 1.95 times the amount of money being paid to all employees over the course of a week in the Current Week.
What is the percentage?The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
Here,
To estimate the reduced Total Wages for the next two weeks combined, we can use the following formula:
Reduced Total Wages = Current Total Wages x (1 - Wage Reduction Percentage)
Reduced Total Wages = C × (1 - 0.025) × 2
Simplifying this expression, we get:
Reduced Total Wages = C × 0.975 × 2
Reduced Total Wages = 1.95C
Therefore, the estimate for the reduced Total Wages for the next two weeks combined would be approximately 1.95 times the amount of money being paid to all employees over the course of a week in the Current Week.
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what is 1/4 x 1/4 x 1/4
Answers
Answer:
1/64
Step-by-step explanation:
1/4 × 1/4 × 1/4
Multiply the fractions.
1/4³
= 1/64
Answer:
The answer is 1/64.
Step-by-step explanation:
1/4 x 1/4 x 1/4 = 1/64
In other ways, you can write this as: 1/4³
Hope this helped!
The original price of a bottle
of dish soap is $2. How much
will Sasha PAY if she buys it
during the sale when it is 50%
off?
Answers
Answer:
$1
Step-by-step explanation:
i hope i read it right
Answer:
1
Step-by-step explanation:
cuz half of 2 is 1
Ana works 25 hours a week at the library. If she is paid $10 an hour, and her net salary each week is $212.50, what percent of her salary is withheld for taxes?
Answers
Approximately 17.65% of Ana's salary is withheld for taxes.
To calculate the percentage of Ana's salary that is withheld for taxes, we need to determine the amount of tax withheld and then express it as a percentage of her net salary.
Number of hours worked per week (H) = 25 hours
Hourly wage (W) = $10
Net salary (S) = $212.50
Calculate the total earnings before taxes:
Total earnings = Hours worked * Hourly wage
Total earnings = 25 hours * $10/hour
Total earnings = $250
Determine the amount of tax withheld:
Tax withheld = Total earnings - Net salary
Tax withheld = $250 - $212.50
Tax withheld = $37.50
Calculate the percentage of salary withheld for taxes:
Percentage withheld = (Tax withheld / Net salary) * 100
Percentage withheld = ($37.50 / $212.50) * 100
Percentage withheld ≈ 17.65%
Therefore, approximately 17.65% of Ana's salary is withheld for taxes.
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What is the quotient of the division below?
612-9-
O 61
O 63
68
O 69
I’ll mark brainlessly
Answers
Answer:
68
Step-by-step explanation:
Hope this helped! :)
A bag of equally sized rhinestones weighs 16 ounces. Each rhinestone weighs 0.016 ounce. If you have 2 bags of rhinestones, how many rhinestones do you have?
Answers
Earth is 92.96 million miles from the su orbits the sun in about 365, 26 days traveling at an average speed of 18.51 miles per secondHow far does Earth travel in minute(60 seconds)
Answers
Answer:
multiply the time and speed the divide the distance with the value of a minute.
Step-by-step explanation:
365 x 26 x 18.52 = 175,754.8 divided by 60 = 2,929.246666666667
simplify into 2,929.2
What values of x
and y
satisfy the system of equations {8x+9y=−36 x+7y=1}
Enter your answer as an ordered pair, like this: (42, 53)
If your answer includes one or more fractions, use the / symbol to separate numerators and denominators. For example, if your answer is (4253,6475),
enter it like this: (42/53, 64/75)
If there is no solution, enter "no"; if there are infinitely many solutions, enter "inf."
Answers
The solution to the system of equations is (x, y) = (-261/47, 44/47) as an ordered pair.
To solve the given system of equations:
Equation 1: 8x + 9y = -36
Equation 2: x + 7y = 1
We can use the method of substitution or elimination to find the values of x and y. Let's use the method of substitution:
From Equation 2, we can solve for x:
x = 1 - 7y
Substituting this value of x into Equation 1:
8(1 - 7y) + 9y = -36
8 - 56y + 9y = -36
-47y = -44
y = 44/47
Substituting the value of y back into Equation 2 to find x:
x + 7(44/47) = 1
x + 308/47 = 1
x = 1 - 308/47
x = (47 - 308)/47
x = -261/47
Therefore, as an ordered pair, the solution to the system of equations is (x, y) = (-261/47, 44/47).
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