Given that for an arithmetic sequence, the first term is q and the second term is qz.

(a) Find the common difference d the general term Tn and the sum of the first n terms. Answer must

be in terms of q and z.
(b) Hence, find the sum of the first 10 terms in terms of q and z. ​

Answers

Answer 1

Answer:

[tex]d = q(z-1)[/tex]

[tex]S_n = \frac{nq}{2}(2 + (n-1)(z-1))[/tex]

[tex]S_{10} = 5q(9z -7)[/tex]

Step-by-step explanation:

Given

[tex]T_1 = q[/tex]

[tex]T_2 = qz[/tex]

Solving (a1): The common difference (d)

d is calculated as

[tex]d = T_2 - T_1[/tex]

This gives:

[tex]d = qz - q[/tex]

Factorize:

[tex]d = q(z-1)[/tex]

Solving (a2): Sum of n terms

This is calculated using:

[tex]S_n = \frac{n}{2}(2*T_1 + (n-1)d)[/tex]

Substitute values for T1 and d

[tex]S_n = \frac{n}{2}(2*q + (n-1)q(z-1))[/tex]

[tex]S_n = \frac{n}{2}(2q + q(n-1)(z-1))[/tex]

Factorize:

[tex]S_n = \frac{n}{2}(q(2 + (n-1)(z-1)))[/tex]

[tex]S_n = \frac{nq}{2}(2 + (n-1)(z-1))[/tex]

Solving (b): Sum of first 10.

In this case, n = 10

So:

[tex]S_n = \frac{nq}{2}(2 + (n-1)(z-1))[/tex]

becomes

[tex]S_{10} = \frac{10 * q}{2}(2 + (10-1)(z-1))[/tex]

[tex]S_{10} = 5 * q(2 + 9(z-1))[/tex]

[tex]S_{10} = 5q(2 + 9(z-1))[/tex]

[tex]S_{10} = 5q(2 + 9z-9)[/tex]

[tex]S_{10} = 5q(2 -9+ 9z)[/tex]

[tex]S_{10} = 5q(-7+ 9z)[/tex]

[tex]S_{10} = 5q(9z -7)[/tex]

Answer 2

Given that the first term and the second term of an arithmetic sequence is q and qz respectively, therefore:

a. Common difference (d) = [tex]\mathbf{d = q(z - 1)}[/tex]

Sum of the first n terms is: [tex]\mathbf{S_n = \frac{qn}{2}[2 + (n - 1)(z - 1)]}[/tex]

b. [tex]\mathbf{S_{10} = 5q(9z - 7)}[/tex]

Recall:

The common difference (d) of an arithmetic sequence = difference between the next term and the previous termSum of n terms, [tex]S_n[/tex] of arithmetic sequence = [tex]\frac{n}{2}[2a + (n - 1)d][/tex]

Given:

First term, [tex]T_1 = q[/tex] (this is also "a")Second term, [tex]\\\\T_2 = qz[/tex]

a. Common difference (d) = [tex]T_2 - T_1[/tex]

Substitute

[tex]d = qz - q\\\\\mathbf{d = q(z - 1)}[/tex]

Sum of n terms:

[tex]T_n = \frac{n}{2}[2a + (n - 1)d][/tex]

Substitute by plugging in the value of d and a

[tex]S_n = \frac{n}{2}[2(q) + (n - 1)q(z - 1)]\\\\S_n = \frac{n}{2}[2q + q(n - 1)(z - 1)]\\\\S_n = \frac{qn}{2}[2 + 1(n - 1)(z - 1)]\\\\\mathbf{S_n = \frac{qn}{2}[2 + (n - 1)(z - 1)]}[/tex]

b. Sum of the first 10 terms:

Substitute n = 10 into  [tex]S_n = \frac{qn}{2}[2 + (n - 1)(z - 1)][/tex]

[tex]S_{10} = \frac{q(10)}{2}[2 + (10 - 1)(z - 1)][/tex]

Simplify

[tex]S_{10} = 5q[2 + (9)(z - 1)]\\\\S_{10} = 5q(2 + 9z - 9)\\\\[/tex]

Add like terms

[tex]\mathbf{S_{10} = 5q(9z - 7)}[/tex]

In summary, given that the first term and the second term of an arithmetic sequence is q and qz respectively, therefore:

a. Common difference (d) = [tex]\mathbf{d = q(z - 1)}[/tex]

Sum of the first n terms is: [tex]\mathbf{S_n = \frac{qn}{2}[2 + (n - 1)(z - 1)]}[/tex]

b. [tex]\mathbf{S_{10} = 5q(9z - 7)}[/tex]

Learn more here:

https://brainly.com/question/14311840


Related Questions

a nursery sells kentucky bluegrass seed for $5.75 per pound and tall fescue seed for $4.50 per pound. the nursery sells a mixture of the two kinds of seed for $5.25 per pound. let k represent the amount of kentucky blue grass seed the nusery uses in in 5 pounds of the mixture

Answers

Answer:

[tex]k=3[/tex]

Step-by-step explanation:

Given that:

Per pound cost of kentucky bluegrass seed = $5.75

Per pound cost of tall fescue seed = $4.50

Per pound cost of mixture of seeds = $5.25

[tex]k[/tex] is the amount of kentucky blue grass.

Total amount of the mixture = 5 pounds

Therefore, amount of tall fescue seed = (5 - [tex]k[/tex]) pounds

To find:

The value of [tex]k[/tex].

Cost of [tex]k[/tex] pounds of kentucky bluegrass seed = [tex]\$5.75k[/tex]

Cost of (5 - [tex]k[/tex]) pounds of tall fescue seed = [tex]\$(5-k)4.50[/tex]

As per question statement:

[tex]5.75k+(5-k)4.50=5.25\times 5\\\Rightarrow 575k+5\times 450 -450k=525\times5\\\Rightarrow 125k=75\times5\\\Rightarrow k =3\ pounds[/tex]

Amount of kentucky blue grass seed used = 3 pounds

Amount of tall fescue seed used = 5-3 = 2 pounds

One pound of candies costs $3.50.

How many pound of candies can be bought for $7?

Answers

Answer: 2 pounds

Step-by-step explanation: 3.50 + 3.50 = 7$ so the answer is 2

Answer:

2 pounds of candy can be bought for $u

Step-by-step explanation:

We have 2 "$3.50" in "$7" and we know that for every $3.50 we have, we can buy 1 pound of candies. Therefore, for $7, we can buy 2 pounds of candies

I hope this helps! :)

Determine the expression

Answers

trinomial and i believe its a degree of 4

What is the value of f(0)?

A. -2
B.-1
C.0
D.1

Answers

Answer:

The answer is C

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

f(0) means the graph of f(x) in which the value of the variable x in the function f is zero.. So, f(0) represents graph for zero value of x

Find all x-intercepts of the function. Express your answer as a list of x-values
f(x) = x^3+x^2+x-1

Answers

Answer:

(0.54368901,0)

Step-by-step explanation:

Subsititute 0 for y and solve to find the x-intercepts.

Choose the definition for the function PLS HELP

Answers

Answer:

Which function please explain

help lol due tonight

Answers

Answer:

Can't see your questions clearly

Step-by-step explanation:

Jill and Katie are making bracelets to sell at The Windmill. They determined that each bracelet would have six beads and four charms.How many bracelets can they sell if they have 80 charms? show all work

Answers

Answer:

20 bracelets

Step-by-step explanation:

Since they must use 4 charms per bracelet, 80 divided by 4 is 20 so 20 bracelets can be made.

romona has $800 in her bank account

Answers

Um not sure what you're asking for bud you didn't finish.

Answer:

Okay?????? I don't think you finished

What is a mutual fund’s

Answers

A pool of money where investors buy shares and a fund manager invests them and gives the investor the return; the way that most Americans invest their money because of the ease. hope that helps!

PLEASE HELP PLEASE i really need help

Answers

Answer:

The constant of proportionality is 6 because 6x2=12 6x4=24 6x6=36 and it keep going

Step-by-step explanation:

which equation is equivalent?

Answers

Answer:

C

Step-by-step explanation:

10x-12
3*2x=6x
3*-4=-12
4x+6x=10x

to calculate the perimeter of the parallelogram you need the two side lengths 12 inches and 15 inches​

Answers

Answer:

the answer would be 54

Step-by-step explanation:

If the shape is a parallelogram then two sides will be equal lengths. same with the other two sides.

if you need the perimeter then you need to add the lengths of all the sides. so that would be 12 12 15 and 15. Add those together and you get 54 inches.

Answer:

i think the answer is 54

Which equation has the correct sign on the product?
A 45(-4)=180 B 13•3= -39
C (-8)(-8)= -64 D (-18)4 = -72​

Answers

answer:

the answer is a

Answer is D because (-18)4 equals -72.... it’s right.

I will give brainliest :)

Write an equation to represent the following situation.

You go to Montgomery's and have $10 to spend.
Drinks cost $2, a bag of chips costs $1, and a pack of
gum costs $0.50. If you buy four bags of chips, how

many drinks can you buy?

•4 + 2x = 10
•2x + 1 = 10
•4 + 2x = 0.50
•1+ 2+ 4 = 10

Answers

Answer:

4 + 2x = 10

x = 3

Step-by-step explanation:

4+2x = 10
so you can buy 3 drinks

You start an account with $2,500 and an annual interest rate of 6.5% compounded yearly. How much is in the account after 7 years?

I need an expression and a solution​

Answers

I'm pretty sure it's this:

Answer:

y=a( 1+  r/n)^ nt

growth is a= 2500     yearly is n= 1

y= 2500( 1+ .065/1)^ 1.7  =   $3884.97

                                 

In March it rained 8.23 inches. It rained 5.96 inches less in April than in March.
How much did it rain in April?

Answers

It rained 2.27 inches in April.

#13 Easton is planning on buying a new pair of cowboy boots. When he
went to Eastern Outfitters last week they costs $60 on the black Friday
sale. This week, they aren't on sale anymore and they cost $85. What is the
percent of change on the boots? *
O 25%
O 45%
O 55%
75%

Answers

It will be 25%

Step-by-step explanation:

4x+c=w make x a subject
plz help me
answer me fast
and i want answer kn step by step

Answers

Answer:

x =w-c/4

Step-by-step explanation:

solution

here,

4x+c=w

or, 4x=w-c

:.x=w- c/x

As x subject, the linear equation is x = 1/4(w -c).

What is linear equation?

Equations whose variables have a power of one are called linear equations. One example with one variable is where ax+b = 0, where a and b are real values and x is the variable.

We have a linear equation,

4x+c=w.

To make x a subject:

4x = w - c

x = 1/4(w -c)

Therefore, the expression is x = 1/4(w -c).

To learn more about the linear equation;

https://brainly.com/question/29739212

#SPJ2

What can be used as a factor to rewrite 16f^2-25g^2?

Answers

Answer:

B = 5g

16f2 - 25g2

Step-by-step explanation:

72.05-29.69?????????????????

Answers

Answer:

42.36

Step-by-step explanation:

Big boi questions just use the caculator

It takes andre 4 minutes to swim 5 laps. At this rate heswims how many laps per minute.

Answers

Answer: It takes him 1 minute and 25 seconds

Step-by-step explanation:

5 divided 4

4 times 1 = 4

5-4=1

put the decimal and make 1=10                           Ans.: 1.25

4 times 2 = 8

10-8=2

put the 0 = 20

4 times 5 =20

20-20=100

Calculate the mean, the variance, and the standard deviation of the following discrete probability distribution. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your final answers to 2 decimal places.) x −41 −27 −8 −3 P(X = x) 0.29 0.35 0.21 0.15

Answers

Answer:

a) Mean is -23.47

b) Variance is 206.59

c) standard deviation is 14.37

Step-by-step explanation:

Given that;

 x              −41        −27        −8         −3

P(X = x)      0.29      0.35      0.21      0.15

X.P            -11.89     -9.45    -1.68     -0.45

a) mean

Mean = ∑X.P = -11.89 +  -9.45 + -1.68 + -0.45

Mean = -23.47

b)  the variance

E(x²) = ∑x²P(X=x)

= (−41²×0.29) + (−27²×0.35) + (−8²×0.21) + (−3²×0.15)

= 487.49 + 255.15 + 13.44 + 1.35

= 757.43

So

Var(X) = E(x²) - [∑X.P]²

Var(X) = 757.43 - [-23.47]²

= 757.43 - 550.8409

= 206.5891 ≈ 206.59

Therefore Variance is 206.59

c) standard deviation

standard deviation SD(X) = √Var(X)

SD(X) = √206.5891

= 14.3732 ≈ 14.37

Therefore standard deviation is 14.37

Find the measures of Angles A,B, and C

Answers

Answer:

A = 102   B = 78   C = 56

Step-by-step explanation:

All three sides of a triangle added up equals 180.

So 41 + 37 + A = 180

A = 102

A straight line = 180 degrees.

One measurement is 102, so the other is 78.

You now know B is 78, and that 3 angles equal 180.

46 + 78 + c = 180

c= 56

If ABC=DEC b=48degree and E=x+4
X=

Answers

♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️

[tex]x + 4 = 48[/tex]

Subtract sides 4

[tex]x +4 - 4 = 48 - 4[/tex]

[tex]x = 44°[/tex]

♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️

The function
The function g(x) has a slope of —5/6. The value of g (x) is 5 when x is 6. Explain how the graph
of g(x) can be used to find the x-intercept. (2 points)
Item

Answers

You have slope and a point so it should be easy.
Set up equation [y-y1= m(x-x1)]
y-5 = -5/6(x-6)
And simplify.
Y=-5/6x +10

Then to find x intercept, plug in a 0 for y
0=-5/6x+10
-10=-5/6x. Subtract 10 from both sides.
-10/ -5/6 = x. Divide by -5/6 on both sides.

X=12

So the x intercept is 12 or (12,0)

There are 24 female performers in a dance recital. The ratio of men to women is 7:8. How many men are in the dance recital?

Answers

I need help to I’m struggling at math

The sum of the number and three times its reciprocal is 79/10. Find the
number.

Answers

hi  

so    x + 3 (-x)  =  79/10

        x-3x = 79/10

         -2x = 79/10

             x =   79/10 / -2  

              X =   79/ 10 *   -1/2

              x =     - 79/20

              x =   - 3.95

For the function below, find ordered pairs for: x = -2 and x = 5 y = 9 - 4x

Answers

Answer:

(- 2, 17 ) and (5, - 11 )

Step-by-step explanation:

Given

y = 9 - 4x

Substitute the given values of x into the equation and evaluate for y

x = - 2, then

y = 9 - 4(- 2) = 9 + 8 = 17 ⇒ (- 2, 17 ) ← ordered pair

x = 5, then

y = 9 - 4(5) = 9 - 20 = - 11 ⇒ (5, - 11 ) ← ordered pair

The graph of an equation with a negative discriminant always has which characteristic?
OO
no x-intercept
no y-intercept
no maximum
no minimum

Answers

No x intercept because there is no solution
Other Questions
3. The area indicated by the blue arrow is a stream. In which direction does this stream flow?Explain. _______________ was the first country that Christopher Columbus approached about funding his trip to find a path to the West Indies.a.Italyb.Spainc.Portugald.England help look at the picture it shows everything A subway car moves at a constant speed of 10 m/s over a period of 10 s. What is the instantaneous speed halfway through this motion Please help me!The temperature of a mixture was 12Celsius at the start of the experiment. After four hours the temperature decreased 25. What is the current temperature of the mixture? Read the following statement: Line segment MN is congruent to line segment OP.Which of the following is an equivalent statement? aMN overbar is similar to OP overbar bMN overbar equals OP overbar cMN overbar is congruent to OP overbar dMN overbar is an element of OP overbar A rental car company is running two specials. Customers can pay $50 to rent a compact carfor the first day plus $6 for each additional day, or they can rent the same car for $20 thefirst day and $12 for every additional day beyond that. Nora notices that, given the numberof additional days she wants to rent the car for, the two specials are equivalent. How manyadditional days does Nora want?Write a system of equations, graph them, and type the solution who was the author of common sense? what did the pamphlet wish to accomplish or what was its purpose? INTRIPLEBDOG FASRENYOU!!!!. HAH AHAH AUBAAGGAB C All of the statements below are false. Change each statement to make it true. Use a pronounto replace the words in bold.1. Tim looks for golf balls in the ocean.He doesn't look for them in the ocean. OR He looks for them in a lake, Helpppppp pleaseeeeeSave me from this questionnnnn The sum of two numbers is 38. The smaller number is 22 less than the larger number. What are the numbers? Jim mowed 5/9 of a lawn. Sarah mowed 1/3 of the lawn. What fraction of the lawn remained to be mowed? what is Mexicos type of government, and the parts of its government. Factor (Hint: Put in Standard Form First)d^2+ 10d + 20 = - 4What is the factored form? anyone have mocks gcse for year11 maths edexcel i need more help i'm so bad at this subject z/4 - 1/2 = 9? pls steps and test Help asap----------------- 7) If x-2 is a factor of P(x)= x^2-5x+a then find the value of a2346please answer Which of the binomials below is a factor of this trinomial? X2 + 8x+16 A. X-4 B. X-8 C. X+4 D. x+8