The recursive formula for the given sequence: 1,5,14,30, . . . is a(n) = a(n-1) + n². It can also be expressed as a(n) = 4a(n-1) -6a(n-2) +4a(n-3) -a(n-4), with a(1) = 1.
A recursive formula of a sequence tells us how to find the nth term, a(n), from the preceding terms, a(n-1), a(n-2), ....
The sequence is:
1,5,14,30, . . . .
Here:
a(1) = 1, a(2) = 5, a(3) = 14, and a(4) = 30.
Let us write:
a(1) = 1
a(2) = 1 + 4 = a(1) + 2²
a(3) = 5 + 9 = 5 + 3² = a(2) + 3²
a(4) = 14 + 16 = 14 + 4² = = a(3) + 4²
So we can conclude the recursive formula is:
a(n) = a(n-1) + n²
However, we can find another recursive formula.
We have:
a(n) = a(n-1) + n² ........ (1)
For n -1 this becomes:
a(n-1) = a(n-2) + (n-1)²
a(n-1) = a(n-2) + n² - 2n + 1 ........ (2)
Subtract (1) - (2):
a(n) - a(n-1) = a(n-1) + n² - a(n-2) - n² + 2n - 1
a(n) - a(n-1) = a(n-1) -a(n-2) + 2n - 1
⇒ a(n) = 2a(n-1) -a(n-2) + 2n - 1 ...(3)
Equation (3) applies for any n, therefore for n -1, eq.(30 becomes:
a(n-1) = 2a(n-2) -a(n-3) + 2(n-1) - 1 ....(4)
Subtract Eq.(3) - Eq.(4).
a(n) - a(n-1) = 2a(n-1) -a(n-2) + 2n - 1 - (2a(n-2) -a(n-3) + 2n - 3)
a(n) - a(n-1) = 2a(n-1) - 3a(n-2) + a(n-3) + 2
⇒ a(n) = 3a(n-1) - 3a(n-2) + a(n-3) + 2 ....(5)
Substitute for n-1 again into Eq.(5):
a(n-1) = 3a(n-2) - 3a(n-3) + a(n-4) + 2 ..... (6)
Subtract Eq.(6) - Eq.(5)
a(n) - a(n-1) = 3a(n-1) -3a(n-2) +a(n-3) +2 -3a(n-2) +3a(n-3) -a(n-4) -2
a(n) - a(n-1) = 3a(n-1) -6a(n-2) +4a(n-3) -a(n-4)
⇒ a(n) = 4a(n-1) -6a(n-2) +4a(n-3) -a(n-4)
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You deposit $2000 and account that earns simple interest an annual rate of 4% how long must you leave the money in the account to warn $500 in interest?
Answer:
time = 6.25 years
Step-by-step explanation:
To calculate the simple interest earned, we can use the following formula:
[tex]\boxed{I = \frac{Prt}{100}}[/tex],
where:
I = simple interest ($500)
P = principal amount ($2000)
r = interest rate (5%)
t = time (? years)
To calculate the amount of time the money must be left, we have to substitute the given values into the equation, and solve for t:
[tex]500 = \frac{2000 \times 4 \times t}{100}[/tex]
⇒ [tex]500 \times 100 = 8000t[/tex] [Multiplying both sides by 100]
⇒ [tex]50000 = 8000t[/tex]
⇒ [tex]t = \frac{50000}{8000}[/tex]
⇒ [tex]t = \bf 6.25 \ \mathrm{\bf {years}}[/tex]
Therefore, the money must be left in the account for 6.25 years to earn $500 interest.
co effiecnet? what is it
The coefficient is defined as a coefficient is a number multiplied by a variable.
We need to define the term coefficient.
What is the coefficient?A quantity or number that is combined with a variable is known as a coefficient. The variable is often multiplied by an integer, which is then printed next to it. It is assumed that the variables with no associated number have a coefficient of 1.
For example, in the expression 6x², 6 is the coefficient of x² but in the expression 2y² + 7, 2 is the coefficient of y².
In terms of a polynomial, a series, or any other expression, a coefficient is, in other words, a multiplicative factor.
Therefore, the coefficient is defined as a coefficient is a number multiplied by a variable.
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Hey i am a student of Grade 10 . due to online classes i havent studied for 2 years and i have no idea about maths. what order do i got to recap everything . some one please help! (CBSE)
Answer:
Khan academy could really help it has videos and lessons so start there.
Step-by-step explanation:
t is a positive whole number.
The expression 2f² + 5t + 2 can never have a value that is a prime number.
Can someone PLEASE explain why.
Since the given expression can be factored in the form (2t+1) and (t+2) so 2t² + 5t + 2 cannot be a prime number.
A whole number greater than 1 that has no factor other than 1 and the number itself is called a prime number.
Some examples of prime numbers are 3,11,19 ... all these numbers can only be divided by 1 or the number itself so they are called prime.
In the given expression 2t² + 5t + 2 , it is said that t is positive whole number. So if t is a positive whole number then the expression can be written as another positive whole number.
Now the expression 2t² + 5t + 2 can be factored into two smaller factors.
2t² + 5t + 2
= 2t² + 4t + t + 2
= 2t (t + 2 )+1 (t+2)
= (t+2) (2t+1)
If t is a positive whole number then t+2 is a whole number and 2t+1 is another positive whole number.
Hence these two numbers are factors of 2t² + 5t + 2 . But a prime number does not have any factors.
So 2t² + 5t + 2 is not a prime number.
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Quadrilateral A B C D is circumscribed about \odot H . Find m .
The value of m is 9 .
What is a Quadrilateral:
A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles. It is formed by joining four non-collinear points. The sum of interior angles of quadrilaterals is always equal to 360 degrees.
What is Circumscribed:
Circumscribed meaning is, a shape is restricted within any other shape. There is no circumvent for the inside figure to pass the outer figure. When we draw a figure around another figure, in such a way, that the outer figure which is surrounding the inside figure touches all its vertices, but doesn’t cross it. The outer figure touches the maximum points of the inside figure without crossing it. These geometrical shapes could be a circle, a square, a triangle, a rectangle or a quadrilateral.
Now,
we know length of tangent from a point to a circle are equal
therefore,
AS=AP,,BP=BQ,CQ=CR,DR=DS
x=BC=BQ+QC=BP+RC=BP+CD−DR=BP+CD−DS
⇒x=5+7−3=9
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At what rate is the amount of flour decreasing?
grams per minute ???
AnAt what rate is the amount of flour decreasing? 270- 180 90- grams per minute 2. As time increases, the amount of flour in the machine increases. Time (minutes)
As the sample size increases, the _____.a. standard error of the mean increasesb. population mean increasesc. standard deviation of the population decreasesd. standard error of the mean decreases
As the sample size increases, the standard error of the mean decreases.
How does the standard error change as the sample size grows?
Standard error diminishes as sample size grows because sample means tend to cluster more closely around the genuine population mean as sample size approaches the true population size.
What grows as the sample size grows?
Our estimate is more accurate since we have more information and hence more data. Our estimate is more certain, our uncertainty is lower, and our precision improves as our sample size grows.Learn more about Standard error
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At a local print shop, 20 copies can be made for $8. At this rate, how how many copies could be made for $32?
80 copies could be made from $32 at a local print shop.
What is termed as the unitary method?The unitary method determines the value of a unit followed by the value of a necessary number of units.Increases or decreases in one quantity will result in a rise or decrease in that other quantity in direct variation. For example, a rise in the amount of goods will result in a higher price.Direct variation is its inverse. When we increase one quantity, the value of some other quantity decreases.For the given question;
$8 is cost for the 20 copies.
Then,
$8 ------> 20 copies.
Divide both side by 8.
$1 -------> 20/8 copies.
For $32, multiply both side by 32.
$32 ------->32×(20/8) copies
$32 -----> 80 copies.
Therefore, 80 copies will be prepared by the amount of $32 at the local print shop.
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The table below shows the price of rolls at a bakery. A baker’s dozen includes 13 rolls for the price of a dozen rolls. How much money is saved by buying a baker’s dozen instead of 13 individual rolls?
Number of rolls
6
8
Baker’s dozen
Price ($)
$3.60
$4.80
$7.20
$0.55
$0.60
$0.70
$0.75
Based on the price to buy a baker's dozen, the money that the baker will save from buying a Baker's dozen instead of 13 individual rolls is $0.60.
What is the difference in cost?The price per roll is:
= 3.60 / 6
= $0.60
The price of 13 individual rolls is:
= 0.60 x 13
= $7.80
The amount saved is:
= Price of 13 rolls - Price of baker's dozen
= 7.80 - 7.20
= $0.60
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Solve each system by elimination.
3a+4b = 9 , -3a - 2b = -3
Answer: 1. D.0.5, 2.5
2. B 10,-1
3. B.-6,-6
4. C 7,5
5. D. Infinite number of solutions
6. A. -1,3
7. C. -1,-0.5
8. B. Seven $1....eight $5...
9. D. 200 muffins
10. A. 2y=1.5x-2 D.-4y+3x=-2
Step-by-step explanation:
i did the quiz, Gl, and thx!
What is the value of the expression below?
12 + |7 - 10|
Answer: The answer is 15
Answer:
15 is the solution.
Step-by-step explanation:
Finding the value of expression,
→ 12 + |7 - 10|
→ 12 + |-3|
→ 12 + 3 = 15
Hence, the value is 15.
Beth ann wants to make brownies for 5 third grade teachers 6 brownies for each teacher if one box makes 7 brownies how many boxes should she buy to have enough
Round the decimal to the nearest whole number. 4.197.
Answer:
4
Step-by-step explanation:
Since 4 is the nearest whole number in 4.197, we need to look at the number next to it.
1 is less than 5, so 4 stays the same.
Find the solution for 4a+3=7-(5-8a)
Answer:
[tex]4a+3=7-5+8a\\4a-8a=7-5-3\\-4a=-1\\\frac{-4a}{-4} =\frac{-1}{-4} \\a=\frac{1}{4}[/tex]
solve
6(x+4)=30 ………………
Answer:
Step-by-step explanation:
6(x+4)=30
So wee need to find out what multiplied by 6 =30
30/6=5
Now wee need to figure out what plus 4 equals 5
5-4=1
1+4=5
6(1+4)=30
So x=1
Hope this helps!^^
1. What is the probability the marble is yellow? The probability it is blue? The
probability
it is red?
Answer:
What is the probability the marble is yellow? = 1/3
The probability it is blue? = 1/3
The probability it is red? = 1/3
Anne and her husband are each starting a saving plan. Deshaun will initially set aside $750 and then add $165 every month to the savings. The amount A (in dollars) saved this way is given by the function A=165N+750 , where N is the number of months he has been saving. Her husband will not set an initial amount aside but will add $395 to the savings every month. The amount B (in dollars) saved using this plan is given by the function B=395N. Let T be total amount (in dollars) saved using both plans combined. Write an equation relating T to N .
The equation relating T to N is T = 560N + 750
What is a function?
A function is defined as an expression or rule that explains or compares the relationship between two variables - the dependent and the independent variable.
From the information given, we have that;
Plan A = 165N + 750Plan B = 395NT is the total amount saved using both plan A and BThis is represented as;
T = A + B
Substitute the values
T = 165N + 750 + 395N
Collect like terms
T = 165N + 395N + 750
Add like terms
T = 560N + 750
Thus, the equation relating T to N is T = 560N + 750
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The cities of Raleigh, Durham, and Chapel Hill, North Carolina, form what is known as the Research Triangle. The approximate latitude and longitude of Raleigh are 35.82°N 78.64°W, of Durham are 35.99^°N 78.91°W , and of Chapel Hill are 35.92°N 79.04°W. Show that the triangle formed by these three cities is scalene.
We have proved that the triangle formed is scalene triangle.
It is given that latitude and longitude of three cities is given.
We have to prove that the triangle formed by the three cities is a scalene triangle.
What is a triangle ?
A triangle is a 3-sided shape. There are three sides and three angles in every triangle, some of which may be the same.
As per the question ;
The Δ formed is a scalene triangle.
Raleigh has latitude and longitude (35.82°N , 78.64°W)
Durham has latitude and longitude (35.99°N , 78.91°W)
Chapel has latitude and longitude (35.92°N , 79.04°W)
Let's represent Raleigh as R , Durham as D and Chapel as C.
Let's check it.
Using the distance formula ;
RD = [tex]\sqrt{(35.99-35.82)^2 + (78.91 - 78.64)^2}[/tex]
RD = [tex]\sqrt{( 0.17^2+ 0.27^2)}[/tex]
RD = [tex]\sqrt{0.102}[/tex]
RD = 0.32
Similarly ;
DC = [tex]\sqrt{0.07^2 + 0.13^2}[/tex]
DC = [tex]\sqrt{0.0218}[/tex]
DC = 0.15
And
RC = [tex]\sqrt{0.10^2 + 0.40^2}[/tex]
RC = 0.41
The sides are all of different lengths, so , we can conclude that this is a scalene triangle.
Thus , the triangle formed by given latitude and longitude of three cities is scalene.
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can 4 x 2/3 be 2/1 ?
Answer:
noStep-by-step explanation:
can 4 x 2/3 be 2/1 ?
2/1 = 2
solve with an equation
4 * 2/3 = 2
8/3 = 2
or
2.66666666667 = 2
the equation is false, so your answer is no
LA and ZB are vertical angles. If mA = (6x - 26)° and m/B = (5x + 6)°,
then find the measure of ZB.
Answer:
∠ B = 166°
Step-by-step explanation:
since ∠ A and ∠ B are vertical angles then they are congruent, that is
∠ A = ∠ B
6x - 26 = 5x + 6 ( subtract 5x from both sides )
x - 26 = 6 ( add 26 to both sides )
x = 32
Then
∠ B = 5x + 6 = 5(32) + 6 = 160 + 6 = 166°
who needs special prayer
Answer: me man
Step-by-step explanation:
holy god
Name two angles that form a linear pair
Answer: [tex]\angle APC[/tex] and [tex]\angle CPB[/tex]
Step-by-step explanation:
See attached image.
Which of the following best describes the ordered pairs listed below? (-3, 2), (3, -10), (5, -14), (10, -24) A. both a relation and a function B. relation only C. function only D. neither a relation nor a function
Since each input is mapped to only one output, the best description of the ordered pairs is given by:
A. both a relation and a function.
When does a relation represents a function?A relation represents a function if each value of the input is mapped to only one value of the output.
For a point in the format (x,y), we have that:
x is the input.y is the output.The meaning is that the input x is mapped to the output y.For this problem, the points are given as follows:
(-3, 2), (3, -10), (5, -14), (10, -24)
Each input is mapped to only one input, that is, there are no repeated values of x, hence the best description of the ordered pairs is given by:
A. both a relation and a function.
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Find the distance AB between the points A(8,2) and B(3,9) round the answer to nearest tenth
The distance between the points is 8.6 units
How to determine the distance between the points?The points are given as:
(8, 2) and (3,9).
The distance is calculated using
d = √(x2 - x1)^2 + (y2 - y1)^2
This gives
d = √(8 -3)^2 + (2 - 9)^2
Evaluate
d = 8.6
Hence, the distance between the points is 8.6 units
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HELP PLEASE Use the substitution method to solve the given system of equations. Write your answer as an ordered pair, (x, y).
y - x = 4
6y + 2x = 4
Answer:
[tex] \sf \large \: x= \frac{−5}{2} \: and \: y = \frac{3}{2} [/tex]
Step-by-step explanation:
Let's solve your system by substitution.
y−x=4;6y+2x=4
Rewrite equations:
−x+y=4;2x+6y=4
Step: Solve−x+y=4for y:
−x+y=4
−x+y+x=4+x(Add x to both sides)
y=x+4
Step: Substitute x+4 for y in 2x+6y=4:
2x+6y=4
2x+6(x+4)=4
8x+24=4(Simplify both sides of the equation)
8x+24+−24=4+−24(Add -24 to both sides)
8x=−20
8x/8 = −20/8
(Divide both sides by 8)
x= −5/2
Step: Substitute −5/2 for x in y=x+4:
y=x+4
y=−5/2+4
y=3/2
(Simplify both sides of the equation)
Emily has a job where she earns $10.25 an hour. If she worked 11.5 hours last week, how much will she earn before taxes are taken out?
100.54
113.33
117.88
121.54
135.67
None of the Above
Answer:
C. $117.88
Step-by-step explanation:
$10.25 per hour
11.5 hours worked
$10.25 * 11.5 = $117.875
round $117.875 to $117.88
C. $117.88
Fill in the blank question.
Find the values of x and y.
(2x+25)+(3x-10)=180
Answer:
x=33
Step-by-step explanation:
Simplifying 2x + 25 + 3x + -10 = 180 Reorder the terms: 25 + -10 + 2x + 3x = 180 Combine like terms: 25 + -10 = 15 15 + 2x + 3x = 180 Combine like terms: 2x + 3x = 5x 15 + 5x = 180 Solving 15 + 5x = 180 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + 5x = 180 + -15 Combine like terms: 15 + -15 = 0 0 + 5x = 180 + -15 5x = 180 + -15 Combine like terms: 180 + -15 = 165 5x = 165 Divide each side by '5'. x = 33 Simplifying x = 33
Answer:
x=33
Step-by-step explanation:
(2x+25)+(3x-10)=180
2x+25+3x-10=180
2x+3x+25-10=180
5x+15=180
5x=180-15
5x=165
5x/5=165/5
therefore x=33
Giving brainlyist HELPP Use the equation one fourth plus s equals 18 over 20 to answer the questions.
Part A: Determine two numbers that the solution to the equation is between. Support your answer using the correct vocabulary. (2 points)
Part B: Solve for the variable. Show your work. (2 points)
Answer:
1/4+s=18/20
Step-by-step explanation:
The answer is between 0 and 1
18/20 = 0.9
1/4+s=0.9
0.9-0.25=0.65
s=0.65
if rectangle a is a 2 by 4 would a 4 by 8 be a scale factor
The required scale factor for the small rectangle is 8.
Given that,
Rectangle a is a 2 by 4 would a be 4 by 8,
To determine the scale factor,
The scale factor is defined as the ratio of modified change in dimensions to original dimensions
Area of the rectangle = product of dimensions = length * width
Area of the former rectangle = 2 * 4
= 8
Area of the later rectangle = 4 * 8 = 32
Scale factor = area of the later rectangle/area of the former rectangle
= 32 / 8
= 8
Thus, the required scale factor for the small rectangle is 8.
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. Find the equation of the line passing through the points (3, 7) and (5, -1). Your answer should be in point -slope form.
y-7=-4(x-3)
find the slope first
-1-7/ 5-3= -8/2
m= -4
put the slope into the point slope formula,
y-y1=m(x-x1)
to get
y-7=-4(x-3)
Answer: m= -4 y=-4x + 19
