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​

Answers

Answer 1
Given:

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.

Answer 2

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 .


Related Questions

what is the square root of 30.3 rounded to the nearest hundreth

Answers

5.50 is the answer:)

Hannah is making cookies. Each batch of cookies uses 3/4 lbs of butter. She is going to make 5 batches. How much butter will she need?

Answers

Answer:3/4*5=3 3/5

Step-by-step explanation:

A plant is already 15.00 meters tall, and it will grow 9 centimeters every month. The plant's height, H (In meters), after x months is given by the following
function.
H(x) = 15.00 +0.09x
What is the plant's height after 30 months

Answers

Answer:

17.7 meters

Step-by-step explanation:

x = 30

15 + 0.09*30

0.09*30 = 2.7

2.7 + 15 = 17.7 meters

What’s 17/5 as a decimal?

Answers

It’s 3.4 because all your doing is dividing 17 into 5

Please help I am so confused!

Answers

Answer:

D

Step-by-step explanation:

If you match what the numbers say you will see the answer is D

For the percent 58%, (a) write a decimal and (b) write a fraction in lowest terms. ​

Answers

Answer:

A) 0.58

B)  58/100

Step-by-step explanation:

Can i have brainliest if its right :)

15-2
6. What is the value of the expression "eight more than three times the difference of four and a number' when
n = 3? (LEVEL 4)

Answers

Answer:

11

Step-by-step explanation:

3(4-n)+8

n=3 so

3(4-3)+8

3(1)+8

3+8

11

solve for x, 3x + 5 > 88

Answers

Answer:

x > 83 / 3

Step-by-step explanation:

Let's solve your inequality step-by-step.

3x+5>88

Step 1: Subtract 5 from both sides.

3x+5−5>88−5

3x>83

Step 2: Divide both sides by 3.

3x/3 > 83/3

x > 83/3

Please help!!
Simplify the expression:
1) (m^2m^3)^4
2) (x^4x)^2

Answers

Hello! I hope i’m correct!! :)

Answer:

1) 20^m

2) x^10

Step-by-step explanation:

[tex](m^{2+3})^{4}[/tex]

[tex](m^{5})^{4}[/tex]

[tex]=[/tex][tex]m^{20}[/tex]

First expression: m^20

______________________________

[tex](x^{5} )^{2}[/tex]

[tex]=x^{10}[/tex]

Second expression: x^10

Hope this helps! If you have any questions about this, feel free to ask! Have a wonderful day! ⭐️

(Sorry if this is incorrect!)

~By, BrainlyAnime~ <3

Answer:

1) 20^m

2) x^10

Step-by-step explanation:

1. Reformatting the input:

Changes made to your input should not affect the solution:

( 1 ) : "m3"  was replaced by "m^3".

m^5 raised to the 4th power = m^( 5 * 4 ) = m^20

_________________________________________________

2.  

x^5 raised to the 2nd power = x^( 5 * 2 ) = x^10

An animal shelter has $2500 in its reserve fund. The shelter charges $40 per animal placement and would like to have at least $4000 in its reserve fund. Write an inequality to represent this situation.

Answers

Answer:

if you search the Question up the fist thing to pop up should say Gr7_Math_Revision_T2. on the 7th page your answer, I think. is there

Step-by-step explanation:

Step 1

Write and solve the inequality:

2,500 + 40a ≥ 4,000, or 40a ≥ 1,500

a ≥ 37.5

Step 2

If the shelter places 30 cats and 10 dogs,

or 40 animals, that will be enough to meet

its goal, because a = 40 is a solution to the

inequality a ≥ 37.5.

Given f(x)=3x2+7 and g(x)=5x-4 solve f(5) btw in the problem the 2 is on top of the X

Answers

Answer:

f(5) = 82

Step-by-step explanation:

To evaluate f(5) substitute x = 5 into f(x) , that is

f(5) = 3(5)² + 7 = 3(25) + 7 = 75 + 7 = 82

How many solutions does this system of equations have?

Answers

one i think, the point where they intersect

Answer:

1 solution

Step-by-step explanation:

A solution to a system of linear equations is where the two lines meet. In this case, the system only intercepts each other once, therefore it only has 1 solution.

Which value of x makes the equation ,3 plus 0.75 ( x minus 2) equals 0.5 ( x + 20 ), true


Someone help fast for the bois??

Answers

Answer:

34

Step-by-step explanation:

3+0.75(x-2)=0.5(x+20)

3+0.75 x-1.5=0.5 x+10

1.5+0.75x=0.5 x+10

0.75x+0.5x=10-1.5

0.25x= 8.5

0.25/25, 8.5/0.25

x=34

how to get 20 dollars by using 25 cents?

please work it out on paper​

Answers

Answer:

below if you cant read it, it takes 80 quarters

Step-by-step explanation:

EMERGENCY! Please give me the correct answer!​

Answers

Answer: 10^4 · 4^5

Since 4^10 and 4^5 have the same base, and it's division, you can subtract the exponents.

(10^4 · 4^10)/4^5 = 10^4 · 4^(10-5) = 10^4 · 4^5

can someone help me with this
2x+4y+3z=6
x-2y+z=-5
-x-3y-2z=-7
solve for x,y,z

Answers

Answer:

x = -7 2/3, y = 1 1/3 and z = 5 1/3.

Step-by-step explanation:

2x+4y+3z=6    .....  1

x-2y+z=-5        ...... 2

-x-3y-2z=-7      .......3

Add equations 2 and 3 to eliminate x:

-5y - z = -12   .....4

Multiply equation 2 by - 2:

-2x + 4y  - 2z = 10

Add this to equation 1:

8y + z = 16   ........ 5

Now add equation 4 to equation 5:

3y = 4

y = 4/3 = 1 1/3.

Now find z by substituting for y in equation 4:

-5(4/3) - z = -12

z = 12 - 20/3

z = 36/3 - 20/3 = 16/3 = 5 1/3.

Finally, we find x by substituting for y and z in equation 1:

2x + 4*4/3 + 3*16/3 = 6

2x = 6 - 16/3 - 16

2x =  18/3 - 16/3 - 48/3 = -46/3

x = 23/3 =  7 2/3.

The table below shows the number of miles four cars can travel on a given amount of gasoline.

Car Mileage

Number of miles traveled
Number of gallons used
Car 1
528
22
Car 2
486
18
Car 3
475
19
Car 4
546
21

Which list shows the cars in order from least miles per gallon to greatest miles per gallon?
HURRY I NEED IT
FIRST RESPONCE GETS BRAINLIEST

Answers

Answer:

the person above me was correct

Step-by-step explanation:

btw it was c

Answer: C

Step-by-step explanation: I took the quiz yesterday

Larry spends a total of 45 minutes running. He begins at a pace of 8 miles per hour and then reduces his speed to 5 miles per hour. Let t represent the number of minutes Larry spent running at the faster pace. Larry models his distance with the expression 8t+5(45−t) to model the situation. Did Larry correctly model the situation

Answers

Answer:

Yes, he did

Step-by-step explanation:

We have that:.

Total time = 45

At time t, he moves at 8mph.

This gives: 8t

The time left is: 45 - t and he moves at 5mph, this gives 5(45 - t).

Hence, the total is

8t + 5(45 - t)

By comparison, we can conclude that Larry modeled the situation correctly.

-1/2x+2=-x+7 i need to know

Answers

[tex] = \frac{ - 1}{ \: \: 2} x + 2 = x + 7[/tex]

[tex] = \frac{ - 1}{ \: \: \: 2} x + 2 - x = 7[/tex]

[tex] = \frac{ - 3}{ \: \: \: 2}x + 2 = 7[/tex]

[tex] = \frac{ - 3}{ \: \ \: 2}x = 7 - 2 [/tex]

[tex] = \frac{ - 3}{ \: \: \: 2} x = 5[/tex]

[tex] x = 5 \: \div \frac{ - 3}{ \: \: \: 2} [/tex]

[tex]x = 5 \times \frac{ \: \: \: 2}{ - 3} [/tex]

[tex]x = \frac{ - 10}{ \: \: \: 3} [/tex]

[tex]∴ \frac{ - 1}{ \: \: \: 2} \times \frac{ - 10}{ \: \: \: 3} = \frac{ - 10}{ \: \: \: 3} + 7

[/tex]

PLEASE PLEASE HELP 5 MINUTES HELPP HELPP WILL MARK BRAINLIEST!!!!!​

Answers

Answer:

Step-by-step explanation:

I & III look good

Find the area of this quadrilateral.

Answers

I think the area is 24

Five times the greater of two consecutive even
integers is 14 greater than three times the lesser
number. What is the greater number?

Answers

Two consecutive even integers:
X, x + 2

5(x+2) = 14 + 3x
5x + 10 = 14 + 3x
2x = 4
X = 2

Greatest is 4

help i need to pass:/

Answers

Answer:

biologists have a 14ft john boat with trolling motor for sale or trade for hipnopotomus

Answer:

false

Step-by-step explanation:

1. Five times a number minus 4 is 6.

Answers

n = a number
5n - 4 = 6

Consider the system of equations in standard form. x + 4y = 26, 3x – 4y = 30 What is the best window range for the x-values to determine the solution? What is the best window range for the y-values to determine the solution? What is the exact solution to the system of equations?

Answers

Answer:  x + 4y = 26 Slope = -0.500/2.000 = -0.250

 x-intercept = 26/1 = 26.00000

 y-intercept = 26/4 = 13/2 = 6.50000

Step-by-step explanation: x + 4y = 26

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is 6.500 and for x=2.000, the value of y is 6.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 6.000 - 6.500 = -0.500 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

   Slope     = -0.500/2.000 = -0.250

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes

x tells us how far along

m is the Slope or Gradient i.e. how steep the line is

b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  x+4y-26  = 0 and calculate its properties

Notice that when x = 0 the value of y is 13/2 so this line "cuts" the y axis at y= 6.50000

 y-intercept = 26/4 = 13/2  =  6.50000

When y = 0 the value of x is 26/1 Our line therefore "cuts" the x axis at x=26.00000

 x-intercept = 26/1  = 26.00000

3x – 4y = 30 Slope = 1.500/2.000 = 0.750

 x-intercept = 30/3 = 10

 y-intercept = 30/-4 = 15/-2 = -7.50000

Notice that when x = 0 the value of y is 15/-2 so this line "cuts" the y axis at y=-7.50000

 y-intercept = 30/-4 = 15/-2  = -7.50000

y = 0 the value of x is 10/1 Our line therefore "cuts" the x axis at x=10.00000

 x-intercept = 30/3  =  10

Answer:

What is the best window range for the x-values to determine the solution?

x-min=-10, x-max = 15

What is the best window range for the y-values to determine the solution?

y-min= 0, y-max = 5

What is the exact solution to the system of equations?

(14, 3)

Can someone please help me ?

Answers

Answer:

The answer is C because 40% of 1450 is 580.

How can you graphically tell if a quadratic equation will have a complex solution?

Answers

Answer:

If it does not cross the x axis.

Step-by-step explanation:

If the quadratic equation does not cross the x axis (or more specifically, if the quadratic is an upward opening parabola and the vertex is above the x axis, or if the quadratic is a downward opening parabola and the vertex is below the x axis), it will have no real solution

The fundamental theorem of algebra says that quadratics must have 2 solutions, so the solutions must be imaginary.

if the equation doesn’t cross the X-axis

Tran planned a rectangular pool and made a scale drawing using centimeters as the unit of measerment he oringinaly planned for the lenth of the pool to be 40m but decided to change to 32 m if the lenth of the pool in his scsale drawing is 8 centimeters which statement about the change of scale is true

Answers

Answer:

The question is not complete, fortunately, I found a match:

Tran planned a rectangular pool and made a scale drawing using centimeters as the unit of measurement. he originally planned for the length of the pool to be 40 m but decided to change it to 32 m. if the length of the pool in his scale drawing is 8 cm, which statement about the change of scale is true?

one cm represented 5 m in the first scale, but now 1 cm represents 4 m in the second scale.

one cm represented 40 m in the first scale, but now 1 cm represents 32 m in the second scale.

one cm represented 1 m in the first scale, but now 1 cm represents 5 ft in the second scale.

one cm represented 4 m in the first scale, but now 1 cm represents 5 m in the second scale.

answer:

one cm represented 5 m in the first scale, but now 1 cm represents 4 m in the second scale.

Step-by-step explanation:

To calculate the scaling factor, we will divide the actual figure by the figure in the scale.

Before the change: 8cm in the drawing represents 40m

8cm ≡ 40m

∴ 1cm ≡ 40 ÷ 8 = 5m

∴ 1 cm represents 5m

After the change: 8cm in the drawing represents 32m

8cm ≡ 32m

∴ 1cm ≡ 32 ÷ 8 = 4

∴ 1cm represents 4m

Therefore, one cm represented 5 m in the first scale, but now 1 cm represents 4 m in the second scale.

Answer:

It’s the first choice

Step-by-step explanation:

b)
2x-5=2(x-1)
Ayuda

Answers

La respuesta es x=2/3 espero que esto ayude :))

Hi I need help :( Pls show your working..Will give brainliest for helpful answer​

Answers

Answer:

number 10

dy/dx =(dy/dt).(dt/dx)

now find

dy/dt=4t-1. , dt/dx=1

so : dy/dx=(4t-1).(1)

dy/dx= 4t-1 ,we have x=t-3-----> t=x+3

subtitute at t

dy/dx= 4(x+3)-1=4x+12-1= 4x+11

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