Answer:
ijbibhibhhhvhj
Step-by-step explanation:
The expression for the sum to 'n' terms of some Arithmetic Sequence are given beliw. Find the 'nth' term of each
i)n^2+2n
ii) n^2- 2n
Arithmetic Sequence
Find:'nth' term of each
i)n^2+2n
ii) n^2- 2n
Solution:1) Sum of first n terms = n² + 2n
We know that,
Sum of first n terms = n/2 * [ a + l ]
Where,
a is first term
l is nth or last term.
Substitute n = 1 to find the sum of first 1 terms i.e., first term (a) of the AP.
→ S₁ = a = (1)² + 2(1)
→ a = 1 + 2
→ a = 3
Hence,
→ n/2 * [ a + l ] = n² + 2n
→ a + l = n(n + 2) * 2/n
→ a + l = 2n + 4
→ l = 2n + 4 - a
→ l = 2n + 4 - 3
→ l = 2n + 1
Hence, the nth term is 2n + 1.
2)S(n) = n² - 2n
→ S₁ = a = (1)² - 2(1)
→ a = - 1
Hence,
→ n/2 * [ - 1 + l ] = n² - 2n
→ l - 1 = n(n - 2) * 2/n
→ l - 1 = 2n - 4
→ l = 2n - 4 + 1
→ l = 2n - 3
Hence, the nth term is 2n - 3.
I hope it will help you.
Regards.
Step-by-step explanation:
Series
The series of a sequence is the sum of the sequence to a certain number of terms. It is often written as Sn. So if the sequence is 2, 4, 6, 8, 10, ... , the sum to 3 terms = S3 = 2 + 4 + 6 = 12.
The Sigma Notation
The Greek capital sigma, written S, is usually used to represent the sum of a sequence. This is best explained using an example:
This means replace the r in the expression by 1 and write down what you get. Then replace r by 2 and write down what you get. Keep doing this until you get to 4, since this is the number above the S. Now add up all of the term that you have written down.
This sum is therefore equal to 3×1 + 3×2 + 3×3 + 3×4 = 3 + 6 + 9 + 12 = 30.
3
S 3r + 2
r = 1
This is equal to:
(3×1 + 2) + (3×2 + 2) + (3×3 + 2) = 24 .
The General Case
n
S Ur
r = 1
This is the general case. For the sequence Ur, this means the sum of the terms obtained by substituting in 1, 2, 3,... up to and including n in turn for r in Ur. In the above example, Ur = 3r + 2 and n = 3.
Arithmetic Progressions
An arithmetic progression is a sequence where each term is a certain number larger than the previous term. The terms in the sequence are said to increase by a common difference, d.
For example: 3, 5, 7, 9, 11, is an arithmetic progression where d = 2. The nth term of this sequence is 2n + 1 .
In general, the nth term of an arithmetic progression, with first term a and common difference d, is: a + (n - 1)d . So for the sequence 3, 5, 7, 9, ... Un = 3 + 2(n - 1) = 2n + 1, which we already knew.
The sum to n terms of an arithmetic progression
This is given by:
Sn = ½ n [ 2a + (n - 1)d ]
You may need to be able to prove this formula. It is derived as follows:
The sum to n terms is given by:
Sn = a + (a + d) + (a + 2d) + … + (a + (n – 1)d) (1)
If we write this out backwards, we get:
Sn = (a + (n – 1)d) + (a + (n – 2)d) + … + a (2)
Now let’s add (1) and (2):
2Sn = [2a + (n – 1)d] + [2a + (n – 1)d] + … + [2a + (n – 1)d]
So Sn = ½ n [2a + (n – 1)d]
Example
Sum the first 20 terms of the sequence: 1, 3, 5, 7, 9, ... (i.e. the first 20 odd numbers).
S20 = ½ (20) [ 2 × 1 + (20 - 1)×2 ]
= 10[ 2 + 19 × 2]
= 10[ 40 ]
= 400
Geometric Progressions
A geometric progression is a sequence where each term is r times larger than the previous term. r is known as the common ratio of the sequence. The nth term of a geometric progression, where a is the first term and r is the common ratio, is:
arn-1
For example, in the following geometric progression, the first term is 1, and the common ratio is 2:
1, 2, 4, 8, 16, ...
The nth term is therefore 2n-1
The sum of a geometric progression
The sum of the first n terms of a geometric progression is:
a(1 - rn )
1 – r
We can prove this as follows:
Sn = a + ar + ar2 + … + arn-1 (1)
Multiplying by r:
rSn = ar + ar2 + … + arn (2)
(1) – (2) gives us:
Sn(1 – r) = a – arn (since all the other terms cancel)
And so we get the formula above if we divide through by 1 – r .
Example
What is the sum of the first 5 terms of the following geometric progression: 2, 4, 8, 16, 32 ?
S5 = 2( 1 - 25)
1 - 2
= 2( 1 - 32)
-1
= 62
The sum to infinity of a geometric progression
In geometric progressions where |r| < 1 (in other words where r is less than 1 and greater than –1), the sum of the sequence as n tends to infinity approaches a value. In other words, if you keep adding together the terms of the sequence forever, you will get a finite value. This value is equal to:
a
1 – r
Example
Find the sum to infinity of the following sequence:
1 , 1 , 1 ,
1
,
1
,
1
, ...
2 4 8 16 32 64
Here, a = 1/2 and r = 1/2
Therefore, the sum to infinity is 0.5/0.5 = 1 .
So every time you add another term to the above sequence, the result gets closer and closer to 1.
Harder Example
The first, second and fifth terms of an arithmetic progression are the first three terms of a geometric progression. The third term of the arithmetic progression is 5. Find the 2 possible values for the fourth term of the geometric progression.
The first term of the arithmetic progression is: a
The second term is: a + d
The fifth term is: a + 4d
So the first three terms of the geometric progression are a, a + d and a + 4d .
In a geometric progression, there is a common ratio. So the ratio of the second term to the first term is equal to the ratio of the third term to the second term. So:
a + d = a + 4d
a a + d
(a + d)(a + d) = a(a + 4d)
a² + 2ad + d² = a² + 4ad
d² - 2ad = 0
d(d - 2a) = 0
therefore d = 0 or d = 2a
The common ratio of the geometric progression, r, is equal to (a + d)/a
Therefore, if d = 0, r = 1
If d = 2a, r = 3a/a = 3
So the common ratio of the geometric progression is either 1 or 3 .
write an equivalent fraction of 7/8 5/12 4/18
Answer:
Step-by-step explanation:
7/8 is 63/72
5/12 is 25/60
4/18 is 8/36
Find the solution to y. For your answer just write the number do not write any symbols or variables. For instance if x = 2, just write 2 in the answer box provided.
-10 + y = 3
Answer:
Step-by-step explanation:
-10 + y = 3 Add 10 to both sides
-10+10+y = 3 + 10 Combine
y = 13
So the answer is 13
You have to do something to get the answer. Algebra is based on logical steps. You don't learn much by seeing 13.
in a class of 36,19 read bioogy,16 read chemistry and 5 were not allowed to read both subjects
Answer:
not hjyftyi uit yfiu hyr k
Step-by-step explanation:
sorry just i need some points
its urgent
Max is x years . Brother is 8 years older , sister is 12 years younger.total age.is 50 years.how old are they ?
Answer:
30
Step-by-step explanation:
12 + 8 = 20 so that means 50-20= 30 so max is 30 years old
Question Progress
Homework Progr
16
0
Work out
5-2x 38
Give your answer as a decimal.
Find the product (10)(-2)(-3)
Answer:
60
Step-by-step explanation:
(10)(-2)(-3) = ?
The minuses cancel out each other
10*2*3 --> 20* 3
Simplified Math
60
Sum
Which is the simplest form of this algebraic expression?
-(6x + 2y) + 4(2x - y)
Answer:
2x-6
Step-by-step explanation:
The first step is to distribute
The second step is to combine like terms and since its an expression you will not have a solution
suppose that there were a positive correlation between the variables u and v. if v is 24 when u is 10, which of these is most likely to be the value of v when u is 20?
Answer: 48
Step-by-step explanation: I got it wrong and it said the correct answer was 48
express 12 1/3 in simplest radical form
Answer:
37/3
Step-by-step explanation:
Convert mixed numbers to improper fraction :
There were 5 boys for every 7 girls at Rayville High School last year. If there were 420 students at the school, how many were girls?
Answer:
240
Step-by-step explanation:
Can you please help me with this?
Answer:
1
hope this helps! :)
How did Washington react to failure?
Tristan bought 23 chicken wings for $57.50. What's the unit cost of one wing?
Answer:
$2.50
Step-by-step explanation:
to find the unit price you would divide it by 23
so 57.50/23=2.50
hope this helps
brainliest?
Rory is at the local mall and her favorite type of jeggings are on sale. The jeggings are now
priced at $7 per pair, tax free. She has $27 with her, but she needs to have $6 leftover to
spend on a movie theater ticket.
Which inequality represents how many pairs of jeggings Rory can buy, where / represents
the number of jeggings?
Answer:
j ≤ 3
Step-by-step explanation:
Since she has $27 with her and needs a balance if $6, the amount of money she can spend on buying the jeggings will be $27-$6 = $21
This means that she has $21 to spend on buying the jeggins
From the question, one jeggings cost $7, so the maximum number of jeggings she can buy is 21/7 = 3
So the inequality that represents this will be;
j ≤ 3
This is because she can decide to buy less than 3, but never can she buy more than 3 because of her budgetary constraints
According to a Wired magazine article, of e-mails that are received are tracked using software that can tell the e-mail sender when, where, and on what type of device the e-mail was opened (Wired magazine website). Suppose we randomly select received e-mails.
a.The expected number of emails is 12
b. The variance and the standard deviation is 7.200 and 2.683
Calculation of expected number of emails, variance and standard deviation:
Given that
P = 0.40 and n = 30
a)
The expected value of received e-mails is
= n × p
= 30 × 0.4
= 12
b)
The variance of emails received is
= n × p × (1 - p)
= 30 × 0.4 × 0.6
= 7.200
Now
The standard deviation of emails received is
= sqrt(variance)
= 2.683
We simply applied the above formula
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−2(−x+4) − 3x+2= −5 what is x?
Answer:
−2(−x+4) − 3x+2= −5 what is x? Well i might be wrong but i think its 2x
Step-by-step explanation:
PEMDAS USE THAT METHOD
Answer: x=-1
Step-by-step explanation:
Which describes the graph of y = (x - 5)2 - 4?
O A. Opens up with a vertex at (5, -4)
B. Opens down with a vertex at (-5,-4)
ting
C. Opens up with a vertex at (-5,-4)
O D. Opens down with a vertex at (5,-4)
Answer:
A. Opens up with a vertex at (5,-4)
Step-by-step explanation:
Apex
Answer: A. Opens up with a vertex at (5,-4)
Step-by-step explanation:
What is the equation of the line that passes through the point (6,-4) and has a
slope of -1/6
Answer:
y = -1/6x-3
hope this helps
Answer:
x+6y+18=0
Step-by-step explanation:
we have to use (y-y1)=m(x-x1)
now,(x1,y1)=(6,-4),m=-1/6
so, (y+4)=-1/6(x-6)
or,6y+24=-x+6
or,x+6y+24-6=0
x+6y+18=0 is the required equation.
:( CAN SOMEONE PLEASE HELP ME
Answer:
I think that Nick is correct because you are going to have to make your answer positive and when you do the two fractions -4/9 + -6/9 are going to be 6/9 + 4/9 and there is already a addition sign in the last problem we are just going to keep it there. And when we add we get 10/9 or 1 1/9
If i get it wrong please don't hate im just in 7 grade ok :(
Answer:
Solution given:
Gobriella is correct.
Nick change - to + in second line.
Which one?
A. B. C. or D?
Answer: D 3sqrt2
Step-by-step explanation:
6x3=18
Sqrt of 18 = 3sqrt2
2. Emily was going to her hotel. It took her 4 hours to get there after work. When she was driving nome she decided to go 15 miles per hour faster. which let her get home 1 hour earlier. How fast did she drive going to the hotel I 3. Give an example of a linear equation that has (a) no solution and ib) infinitely many solutions
Answer:
She could not even fit in the hotel in the first place
Step-by-step explanation:
find the x and why intercepts of the graph of the equation x + 2y = 10
NEED HELP ASAP
Answer:
Step-by-step explanation:
To find the x-intercept and y-intercept of the equation we need to understand what is an intercept. Intercept means to intersect/cut so x-intercept means the equation cuts the x-axis and y-intercept means the equation cuts the y-axis.
So if the equation cuts the x-axis the coordinate should be [tex](x,0)[/tex] and if the equation cuts the y-axis the coordinate should be [tex](0,y)[/tex] , and since the equation satisfies these both points in other words these two points lie on the line they can be put into the equation. So here we go,
For x-intercept
[tex]x+2y=10\\x+2(0)=10\\x=10[/tex]
so the x-intercept is (10,0)
For y-intercept
[tex]x+2y=10\\0+2y=10\\2y=10\\y=10/2\\y=5[/tex]
so the y-intercept is (0,5)
I also have attached an example of how intercepts looks like you can take a look i plotted the graph an you can see our answers check out.
What value is missing from the table?
Answer:
12
Step-by-step explanation:
15/1.25 is 12
Answer:
12
Step-by-step explanation:
What is the value of g?
25(g−7)=3
Answer: G=7.12
Step-by-step explanation: 25g-175=3 add 175 to both side which makes the equation 25g=178 and divide 178 by 25 which gives you 7.12
Answer:
g=178/25
Step-by-step explanation:
Step 1: 25(g-7)/25=3/25
Step2: g-7=3/25
Step 3: g-7+7=3/25 + 7
Answer: g=178/25
Decimal form would be g=7.12
I need major help !!
Answer:
44 degrees
Step-by-step explanation:
The sides are even so there gonna be the same all around...please give brainlist!
8/3 × (−2 1/2) = ?
Thanks!
Answer: -20/3
Step-by-step explanation:
Answer: It equals -8/3
question in picture.
Answer:
4,3,0.5 youre welcome, good Luckk : )
During this basketball season trey took 250 shots and made 72 shots what was his shooting percent
Caleb is a junior at JA Enterprise High School and plays rugby on the school team. He has a part-time job at a local fast-food restaurant and works 20 hours a week for $7.60 an hour. His gross monthly income is $608.00, and his take-home pay is $501.64. In his spare time, Caleb walks a dog belonging to an elderly neighbor. He earns an additional $40.00 per month for that chore.
Caleb's parents gave him a used car, and he pays his own insurance, which is $110 per month. His gas and maintenance expenses usually run $65 a month. He also has a cell phone and pays his parents $30 each month for his share of the family plan.
In three weeks, Caleb plans to take a day trip to Six Flags amusement park. His admission will cost $50, and he plans to have $35 for meals and another $50 for incidentals.
Caleb needs to start saving for Christmas, which is in four months, and he usually spends around $200 on gifts for his family and friends. He also would like to buy a new $80 e-reader this month.
Though Caleb's parents will help pay for his college tuition, he knows that in two years, he will have to pay for his books and earn his own spending money. He is estimating that each semester, he will spend $300 on books and $600 on personal needs.
How much are Caleb's regular monthly expenses?
Answer:
Caleb's net monthly income would be $541.64
Step-by-step explanation:
His monthly take home pay from restaurant is $501.64
Since he earns additional $40 per month for dog walk,
Therefore, his monthly pay would be ;
= $501.64 + $40
Caleb's net monthly income =$541.64
Caleb's monthy regular expenses including car insurance and books is
$355.
What is an equation?An equation is written in the form of variables and constants separated by the operation of multiplication and division,
An equation states that terms in different forms on both sides of the equality sign are equal.
Multiplication and division do not separate the terms of an equation.
From the given information, The total net earnings of Caleb is,
= $(501.64 + 40).
= $541.64.
Now, The expenses for Car insurance, Gas, and cell phone recharge is,
= $(110 + 65 + 30).
= $205.
Again, For books and personal needs, he have to spend $(300 + 600).
= $900.
Monthly it is,
= 900/6.
= $150.
Therefore, Caleb's regular expenses is,
= $(205 + 150).
= $355.
learn more about equations here :
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