a3 = 81 and a6 =648 what is the 1st term?
Answers
The answer depends on what you know about the sequence \(a_n\)...
If \(a_n\) is geometric, then
\(a_n=ra_{n-1}\)
for some fixed number r. Using this recursive rule, we get
\(a_{n-1}=ra_{n-2}\implies a_n=r^2a_{n-2}\)
\(a_{n-2}=ra_{n-3}\implies a_n=r^3a_{n-3}\)
and so on, down to the 1st term in the sequence:
\(a_n=r^{n-1}a_1\)
This means
\(a_3=81=r^2a_1\)
\(a_6=648=r^5a_1\)
so that
\(\dfrac{648}{81}=\dfrac{r^5a_1}{r^2a_1}\implies 8=r^3\implies r=2\)
Now solve for \(a_1\):
\(81=2^2a_1\implies a_1=\boxed{\dfrac{81}4}\)
If instead \(a_n\) is arithmetic, then
\(a_n=a_{n-1}+d\)
for some fixed number d. Similarly, we can get the n-th number in the sequence in terms of the 1st one:
\(a_{n-1}=a_{n-2}+d\implies a_n=a_{n-2}+2d\)
\(a_{n-2}=a_{n-3}+d\implies a_n=a_{n-3}+3d\)
and so on, down to
\(a_n=a_1+(n-1)d\)
Then
\(a_3=81=a_1+2d\)
\(a_6=648=a_1+5d\)
Solve for d :
\((a_1+5d)-(a_1+2d)=648-81\implies 3d=567\implies d=189\)
Now solve for \(a_1\):
\(81=a_1+2\cdot189\implies a_1=\boxed{-297}\)
Related Questions
d) 81 is 45% of what number
Answers
Answer:
180
Step-by-step explanation:
\(0.45x = 81\\45x = 8100\\x = 180\)
Mark has, in his kitchen, the ingredients needed to prepare 78 different recipes with two ingredients
each. Of those 78 different recipes, there are no two recipes that share the exact same two ingredients.
How many ingredients does he have?
Answers
Answer:
13 ingredients
Step-by-step explanation:
The number of different recipes that can be made from 2 ingredients = 78
Since there are no two recipes that share the exact same two ingredients ;
The number of recipes in Mark's kitchen is:
nCk = 78
k = Number of ingredients per recipe = 2
n = total number of ingredients in kitchen
Total recipes made = 78
n! / (n-r)!r! = 78
n! / (n - 2)!2! = 78
n(n-1)(n-2)! / (n-2)! * 2 * 1 = 78
n(n - 1) / 2 = 78
n(n-1) = 156
n² - n = 156
n² - n - 156 = 0
Factorize :
n² - 13n + 12n - 156 = 0
n(n - 13) +12(n-13) = 0
(n-13) = 0 ; (n+12) = 0
n = 13 or n = - 12
n cannot be negative
Hence, n = 13
Number of ingredients is 13
What does it mean to simplify???
Answers
Example:
2/4 = 1/2; Fraction was simplified to make it simpler
4x = 8
x = 2
The equation was simplified to simplest terms of 1x.
Renee is making a scale diagram of her MP3 player. The length of her scale drawing is 8 inches, and the width is 14 inches. The actual length of the MP3 player is 4 centimeters, and the width is 7 centimeters. This is , and the scale factor is .
Answers
Answer:
2
Step-by-step explanation:
Scale Factor = \(\frac{AnySideOfDiagram}{AnySideOfMP3Player}\)
So,
Scale Factor = \(\frac{8}{4} = \frac{14}{7}\) = 2
So,
The scale factor is 2
A hamburger weighs 1/3 pound, and an
order of french fries weighs 1/4 & pound. How
many pounds total will a meal of hamburger
and french fries weigh?
The meal will weigh 'pounds.
Answers
Answer:
I have no idea
Step-by-step explanation:
1/3 + 1/4
(1 x 4) + (1 x 3)
____________
3 x 4
= 4 + 3
________
12
7/12
Answer:
7/12 pounds
Step-by-step explanation:
1/3 * 1 = 1/3
1 = 4/4
then
1/3 * 4/4 = 1/3
1/3 * 4/4 = (1*4)/(3*4) = 4/12
1/3 = 4/1|2
1/4 * 3/3 = (1*3) / (4*3) = 3/12
1/4 = 3/12
then:
1/3 + 1/4 = 4/12 + 3/12 = (4+3)/12 = 7/12
The meal will weigh 7/12 pounds.
Solve the compound inequality.
4x-5<-2 or x+1>3
Answers
The solution of the compound inequality is x < 3/4 or x >2.
What is inequality?In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to each other. To solve the inequality, you may multiply or divide each side by the same positive number, add the same amount to each side, take the same amount away from each side, and more. You must flip the inequality sign if you multiply or divide either side by a negative number.
Given:
A compound inequality:
4x - 5 < -2 or x + 1 > 3.
Solving inequality,
4x < -2 + 5 or x > 2
4x < 3 or x > 2.
x < 3/4 or x >2.
Therefore, the solution is x < 3/4 or x >2.
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Which is a unit rate?
Please help me
Answers
Question↓
Which is a unit rate?
Answer↓
The Correct Answer Is 6 hours: 1 Book
unit rate. • a comparison of two measurements in which. one of the terms has a value of 1.
Hope this helps!! ^ω^
xXxAnimexXx
What is the measure of 23, in degrees, in the figure shown? 30° 137° 3
Answers
Answer:
73°
Step-by-step explanation:
This equation uses two supplementary angles. Supplementary angles are two angles that add up to 180 degrees.
The triangle has two angles implied, 30 degrees and (180-137) degrees.
(180 - 137) = 43 degrees
Now that we have two angles inside the triangle, we subtract them from 180 to find the last angle.
180 - 43 - 30 = 107
The angle with a measure of 107 degrees and angle 3 are supplementary
180 - 107 = 73 degrees
Answer:73.4° Fahrenheit.
Step-by-step explanation:
Match each expression with A, B, C or D.
A=a^3
B=6a
C=12a
D=3a^2
i)3a x 4
ii)a^2xa
iii) 6 1/2 a^2
Answers
The matching expressions are:
\(i) 3a x 4 = C (12a)\\ii) a^2 x a = A (a^3)\\iii) 6 × 1/2 a^2 = D (3a^2)\)
i) 3a x 4 can be represented as C (12a) since multiplying 3a by 4 gives 12a.
ii) a^2 x a can be represented as A (a^3) since multiplying a^2 by a gives a^3.
iii) \(6 \times 1/2 a^2\) can be represented as D (3a^2) since multiplying 6 by 1/2 and then by a^2 gives 3a^2.
To understand the matching expressions, let's break down each one:
i) 3a x 4:
This expression represents multiplying a variable, 'a', by a constant, 4. The result is 12a, which matches with C (12a).
ii) a^2 x a:
This expression represents multiplying the square of a variable, 'a', by 'a' itself. This results in a^3, which matches with A (a^3).
iii) 6 × 1/2 a^2:
This expression involves multiplying a constant, 6, by a fraction, 1/2, and then multiplying it by the square of 'a', a^2. The final result is 3a^2, which matches with D (3a^2).
Therefore, the matching expressions are:
i) 3a x 4 = C (12a)
ii) a^2 x a = A (a^3)
iii) 6 × 1/2 a^2 = D (3a^2)
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This is a gardener's floor plan of their backyard. Enter the amount of grass, in square yards
needed to completely cover the backyard.
Answers
Step-by-step explanation:
that is a tiny, tiny backyard ...
what I see immediately is that the area is a combination of multiple simple figures :
1 square on top,
1 smaller rectangle in the middle,
1 larger rectangle at the bottom.
we only need to calculate the areas of all 3 figures and then add then up for the total.
that's it.
the top square is easy :
it is a 3cm×3cm square, so the area is 9 cm²
the bottom rectangle is also easy :
it is a 10cm×3cm rectangle, so the area is 30 cm²
for the dimensions of the middle rectangle we need to think a little bit :
the width (or "height") is clear : 3 cm.
what is the length ?
the answer is given by some other pieces of information in the diagram. look closely : is the length of the middle rectangle not the same as the sum of the side length of the top square (3 cm) and the top side segment left of the top square : 2 cm ?
yes, it is, so the length is 3+2 = 5 cm.
and the middle rectangle is 5×3 = 15 cm².
so, the total area is
9 + 30 + 15 = 54 cm²
this is a backyard for Barbie and Ken ...
Amount of grass in the backyard is 0.641 square yards
What is a square?
Any 2-dimensional figure bounded by 4 sides whose all sides are equal and all the angles are 90° is called a square.
How to find the area of a square?Area of a square can be found by squaring the length of one of its sides.
What is a rectangle?Any 2-dimensional figure bounded by 4 sides whose opposites sides are equal and parallel and all the angles are 90° is called a rectangle.
How to find out area of a rectangle?Area of a rectangle can be found by multiplying its length by its breadth.
How to calculate the area in the given problem?The given figure is a combination of multiple figures.
1 square of length 3 cm on top1 rectangle in the middle whose length is ( 2 + 3) cm = 5 cm, and breadth is 3 cm1 larger rectangle at the bottom whose length is 10 cm and breadth is 3 cm.We will have to calculate these areas and sum up the areas to get the entire area.Area of the square at the top = ( 3 x 3) cm² = 9 cm²Area of the middle rectangle = ( 5 x 3) cm² = 15 cm²Area of the bottom rectangle = (10 x 3) cm² = 30 cm²∴ Total area of the backyard = ( 9 + 15 + 30) cm² = 54 cm²
Amount of grass in the backyard = (54 x 0.109² ) square yards
= 0.641 square yards ( 1 cm = 0.109 yards)
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A Father is three times as old as his son. Five years later he will be only 2 ½ times older than his son: Find their present age
Answers
Answer: the present age of son is 15 years, and the present age of father is 45 years.
Step-by-step explanation:
Let the age of son be 'x' and that of his father be 'y'.
So according to the question, the 1st equation that can be formed will be:
y = 3(x) ->(1)
[as father's age is 3 times the age of son]
Now after 5 years:
Age of son = (x+5)
Age of father = (y+5)
So according to the question, the 2nd equation that can be formed will be:
(y+5) = 2.5×(x+5) ->(2)
[as father is now two and half times old as his son]
Substituting the value of equation (1) in (2), we get:
(3x+5) = 2.5×(x+5)
=> 3x + 5 = 2.5x + 12.5
=> 3x - 2.5x = 12.5 - 5
=> 0.5x = 7.5
=> x = 7.5/0.5 = 15
Thus, the present age of son is 15 years.
Note : we can calculate the age of father by putting the value of x in equation (1) or (2):
y = 3(x) = 3 × 15 = 45 (putting x in equation 1)
Thus, the present age of father is 45 years.
what is the volume explain why will mark branniest GIVE ME EXPLANATION
Answers
Answer:
63 yd³
Step-by-step explanation:
V = lwh
V = 7 × 3 × 9
V = 189
Since the cubes are 1/3 yd³, you now multiply the volume of this figure which is if it were a 1 yd³.
189 × \(\frac{1}{3}\)
63 yd³
Find 6/11/-3. Write in simplest form
Answers
Answer:
Bit confusing but I'll try my best.
Step-by-step explanation:
6/11 diving by -3.
-2/11.
Evaluate v(x) = 14 – 3x when x = -3,0, and 4.
v(-3) =
v(0) =
v(4) =
Answers
\(v(x) = 14 - 3x\\v(-3) = 14 - 3 \cdot (-3) = 14 + 9 = 23\\v(0) = 14 - 3 \cdot 0 = 14 - 0 = 14\\v(4) = 14 - 3 \cdot 4 = 14 - 12 = 2\)
What is the volume of a cube with
1
inch sides?
2
? ?
?
?
ins
* in.
2 in.
Ź
2 in.
SUSAR CUBES
Answers
Answer:
4
Step-by-step explanation:
volume - side x side
2 x 2 - 4
X + 2x + 8 = X = 10 + 6
Answers
Answer:
x = 8/3
Decimal form: 2.66666666667
Step-by-step explanation:
Given:
\(x+2x+8=10+6\)
Solution:
\(\mathrm{Add\:similar\:terms:}\:x+2x=3\)
\(3x+8=10+6\)
\(\mathrm{Subtract\:}8\mathrm{\:from\:both\:sides}\)
\(3x+8-8=16-8\)
\(\mathrm{Simplify}\)
\(3x=8\)
\(\mathrm{Divide\:both\:sides\:by\:}3\)
\(\frac{3x}{3}=\frac{8}{3}\)
\(\mathrm{Simplify}\)
\(x=\frac{8}{3}\)
\(x=2.66666666667\)
~lenvy~
Answer/Step-by-step explanation:
Base on the given we are..
Solving for x:
Solve:
X + 2x + 8 = X = 10 + 6 {x+2x= 3x}
3x + 8 = 10 + 6 {10+6=16}
3x + 8-8=16-8 {16-8=8}
3x = 8 {divide both sides by same term}
3x/3 = 8/3
Therefore,
x = 8/3
Decimal form = 2.66666666667
[RevyBreeze]
Hello everyone, I hope you had a great day, I need help on this please, thanks.
Answers
Answer:
A
Step-by-step explanation:
A. is the only correct solution to this question because you can also see that if we take 8 and divide this by 32 we get 1/4, so we know it is true.
Because 8 is the same as 8/1 -> 8/1 x 1/4 is 8/4 or 2
Please mark brainiest :))
1. Prove ~ (Pvq) <=> (~P^~9).
Algebra of propositional variables
2. P^ q = q^q
P v q = q v p
show that both are commutative
Answers
Based on the information, both ∧ (conjunction) and ∨ (disjunction) satisfy the commutative property.
How to explain the commutative property.It should be noted that to prove the commutativity of the logical connectives ∧ (conjunction) and ∨ (disjunction), we need to show that they satisfy the commutative property
From the truth table, both P ∧ Q and Q ∧ P have the same truth values for all combinations of truth values of P and Q. Therefore, we can conclude that P ∧ Q ≡ Q ∧ P, and ∧ (conjunction) is commutative.
In order to prove P ∨ Q ≡ Q ∨ P, we construct a truth table for both expressions. Both P ∨ Q and Q ∨ P have the same truth values for all combinations of truth values of P and Q. Therefore, we can conclude that P ∨ Q ≡ Q ∨ P, and ∨ (disjunction) is commutative.
Hence, both ∧ (conjunction) and ∨ (disjunction) satisfy the commutative property.
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A customer wants rhombus-shaped tiles for his countertops. The height is 3 inches and the length of each side is 6.5 inches. What is the area of each tile?
Answers
Answer:
The length of each side of the tiles is 7.2 in
Step-by-step explanation:
Given:
Height of the tile, h = 3 in
Area of each tile, A = 21.6 in²
Length of each side, a = ?
We know:
A rhombus is a type of quadrilateral whose opposite sides are parallel and equal.Also, the opposite angles of a rhombus are equal and the diagonal intersect each other perpendicularly.
To find the area of a rhombus when the measures of its height and side length are given, use the formula
A = Base X height
21.6 in² = a X 3 in
a = 7.2 in
Therefore, the length of each side of the tiles is 7.2 in
2. The table represents equivalent ratios. What is the missing value of x in the table? (1 point)
xy
8 4
? 3
42
2 1
9
7
6
5
Answers
There is no proportional relationship between x and y.
No, all ratios yx are not equivalent.
We have,
"Two or more number or variable are said to be in proportion if the ratio between them are equivalent to each other."
According to the question,
Given x : 8 , 10, 12, 14
y : 5, 7, 9, 11
Ratio between different values of x and y are not equivalent
( 8 /5) ≠ (10 / 7)≠ (12 /9) ≠(14 /11)
We conclude there is no proportional relationship between x and y.
Hence, Option(1) is the correct answer.
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complete question:
Is there a proportional relationship between x and y? Explain.
x 8 10 12 14
y 5 7 9 11
1. No; all ratios yx are not equivalent.
2.Yes; each x value and y value increases by 2.
3.Yes; the difference between each x value and y value is 3.
4.No; all ratios xy are greater than 1.
Use the graphing tool to see what happens to the square root function when you change more than one parameter at a time. In what ways is this new graph different than the parent graph of ? Write down your observations for each of these transformed functions. Then try a few of your own.
Answers
The graphs are different because they are transformed from the parent functions f(x) differently
How to determine how the graphs are differentFrom the question, we have the following parameters that can be used in our computation:
Parent function, f(x) = x
Also, we have: g(x) = (x - 2) + 1, g(x) = 3x - 6, g(x) = 2(x + 3) + 10 and g(x) = (6x - 1)
The equations listed below represent transformations of the parent graph of f(x) = x:
g(x) = (x - 2) + 1
The graph of g(x) is obtained by shifting the parent graph of f(x) = x to the right by 2 units and up by 1 unit.
g(x) = 3x - 6
The graph of g(x) is obtained by vertically stretching the parent graph of f(x) = x by a factor of 3 and then shifting it downward by 6 units.
g(x) = 2(x + 3) + 10
The graph of g(x) is obtained by shifting the parent graph of f(x) = x to the left by 3 units, vertically stretching it by a factor of 2, and then shifting it upward by 10 units.
g(x) = (6x - 1)
The graph of g(x) is obtained by stretching the parent graph of f(x) = x horizontally by a factor of 1/6 and then shifting it downward by 1 unit.
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Complete question
Use the graphing tool to see what happens to the functions when you change more than one parameter at a time. In what ways is this new graph different than the parent graph of f(x)
Write down your observations for each of these transformed functions, then try a few of your own.
g(x) = (x - 2) + 1
g(x) = 3x4 - 6
g(x) = 2(x + 3) + 10
g(x) = (6x - 1)
PLS HELP!
Adrian is going to invest $9,900 and leave it in an account for 10 years. Assuming the interest is compounded continuously, what interest rate, to the nearest hundredth of a percent, would be required in order for Adrian to end up with $17,700?
Answers
Answer:
5.81%
Step-by-step explanation:
Please help ASAP I am in desperate need
Answers
Answer:
x = 9
TH = 12
Step-by-step explanation:
since TM is half of HM, the equation is:
2(21 - x) = 3x - 3
42 - 2x = 3x - 3
45 = 5x
x = 9
TH = 21 - x
TH = 21 - 9
TH = 12
Determine the validity of the converse and give a counterexample if the converse is not valid.
If it is sunny, then it is 80° Fahrenheit. The converse is valid.
If it is 80° Fahrenheit, then it is sunny. The converse is valid.
If it is not sunny, then it is not 80° Fahrenheit. The converse is invalid; a counterexample is a day that is not 80° Fahrenheit and not sunny.
If it is 80° Fahrenheit, then it is sunny; The converse is invalid; a counterexample is a day that is 80° and cloudy.
Answers
Your analysis is correct. Here's a summary:
If it is sunny, then it is 80° Fahrenheit. (Original statement)
Converse: If it is 80° Fahrenheit, then it is sunny. (Valid)
If it is 80° Fahrenheit, then it is sunny. (Original statement)
Converse: If it is sunny, then it is 80° Fahrenheit. (Valid)
If it is not sunny, then it is not 80° Fahrenheit. (Original statement)
Converse: If it is not 80° Fahrenheit, then it is not sunny. (Invalid)
Counterexample: A day that is not 80° Fahrenheit and not sunny (e.g., 70° and cloudy).
If it is 80° Fahrenheit, then it is sunny. (Original statement)
Converse: If it is sunny, then it is 80° Fahrenheit. (Invalid)
Counterexample: A day that is 80° Fahrenheit and cloudy.
In cases 1 and 2, the original statements and their converses are valid because the relationship between "sunny" and "80° Fahrenheit" holds in both directions. However, in cases 3 and 4, the converses are invalid because there are counterexamples where the second part of the statement (either "not 80° Fahrenheit" or "cloudy") does not necessarily imply the first part ("not sunny" or "80° Fahrenheit").
Suppose the only measuring cup you own is a one-third measuring cup. Using this measuring cup how many scoops will you need to have 3 2/3 cups of sugar
Answers
Answer:
11
Step-by-step explanation:
for each full cup of sugar, you need 3 one-third measuring cups.
we have 3 full cups so 3×3=9
and another 2 for the 2/3, 9+2=11
so a total of 11 scoops
Rearrange 2x = y/w to make w the subject
Answers
What is the sum of the interior angle
measures of a regular hexagon? Show
or explain how you can use the sum
of the interior angles of a triangle to
determine the answer.
Answers
Answer:
The sum of the interior angle measures of a regular hexagon is 720°.
Step-by-step explanation:
To find the sum of the interior angle measures of a regular hexagon, we can use the fact that any polygon can be divided into triangles. The formula to find the sum of the interior angles of any polygon with n sides is:
Sum of interior angles = (n - 2) × 180°
In the case of a hexagon, n = 6. So, we can plug this value into the formula:
Sum of interior angles = (6 - 2) × 180° = 4 × 180° = 720°
The sum of the interior angle measures of a regular hexagon is 720°.
To explain this using triangles, let's divide the hexagon into triangles. A hexagon can be divided into four triangles by drawing three diagonals from one vertex to the three non-adjacent vertices. Since the sum of the interior angles of each triangle is 180°, and we have four triangles:
Sum of interior angles of hexagon = 4 × 180° = 720°
The sum of the interior angle measures of a regular hexagon is 720°.
there were 3 ponies. rodrick rode each pony 6 times. how many pony rides did sarah take?
Answers
Answer: 0
Step-by-step explanation: I am going by the words , there WERE 3 ponies. So Sarah couldn’t take any rides.
Help pls will give BRAINLIST
Answers
Hope this helps!
Annual sales of a certain brand of phone are expected to grow at the rate off' (t) = 0.18t^2+ 0.16t+ 2.64million phones per year. Use a definite integral to determine the total number of phones that will be sold over the next 10 years.
Answers
Given the Annual sales of a certain brand of phones are expected to grow at the rate of:
\(f^{\prime}(t)=0.18t^2+0.16t+2.64\)We will use the definite integral to determine the total number of phones that will be sold over the next 10 years.
\(\begin{gathered} \int f^{\prime}(t)=\int ^{10}_00.18t^2+0.16t+2.64 \\ \\ =(\frac{0.18}{3}t^3+\frac{0.16}{2}t^2+2.64t)|^{10} \\ \\ =0.06\cdot10^3+0.08\cdot10^2+2.64\cdot10 \\ =94.4 \end{gathered}\)So, the answer will be the number of phones = 94.4 million phones