MATHEMATICS HIGH SCHOOL

The sum of the first 10 terms of an arithmetic sequence is 235 and thesum of the second 10 terms is 735. Find the first term and the common

difference.

Answers

Answer 1
Answer:

If the sum of the first 10 terms of an arithmetic sequence is 235 and the sum of the second 10 terms is 735. The first term of the arithmetic sequence is -21.5 and the common difference is 10.

What is the arithmetic sequence?

Let's a represent the first term of the arithmetic sequence and the common difference as d

Formula for the sum of the first n terms of an arithmetic sequence is given by:

S_n = (n/2) * [2a + (n - 1)d]

So,  

Sum of the first 10 terms: S₁₀ = 235

Sum of the second 10 terms: S₂₀ = 735

First 10 terms:

S₁₀ = (10/2) * [2a + (10 - 1)d]

235 = 5 * [2a + 9d]

47 = 2a + 9d

Second 10 terms:

S₂₀ = (10/2) * [2a + (20 - 1)d]

735 = 5 * [2a + 19d]

147 = 2a + 19d

Now we have a system of equations:

2a + 9d = 47

2a + 19d = 147

(2a + 19d) - (2a + 9d) = 147 - 47

10d = 100

d = 10

Substitute it into the first equation to solve for "a":

2a + 9(10) = 47

2a + 90 = 47

2a = -43

a = -21.5

Therefore the first term of the arithmetic sequence is -21.5 and the common difference is 10.

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Answer 2
Answer:

Answer:

             \bold{a_1 = -42.65}\n\n\bold{d=14.7}

Step-by-step explanation:

The sum of the first 10 terms of an arithmetic sequence is:

S_(10)=a_1+a_2+...+a_(10)=(a_1+a_(10))/(2)\cdot10=(2a_1+(10-1)d)/(2)\cdot10

(2a_1+(10-1)d)/(2)\cdot10=235\n\n(2a_1+9d)\cdot5=235\n\n2a_2+9d=47

the  sum of the second 10 terms is:  a₁₁ + a₁₂+...+ a₂₀

And the sum of the first 20 terms of an arithmetic sequence is:

S_(20)=a_1+a_2+...+a_(10)+a_(11)+...+a_(20)=(2a_1+(20-1)d)/(2)\cdot10

so the  sum of the second 10 terms is:

a_(11)+a_(12)+...+a_(20)=S_(20)-S_(10)

Therefore we have:

(2a_1+(20-1)d)/(2)\cdot10-(2a_1+(10-1)d)/(2)\cdot10=735\n\n(2a_1+19d)\cdot5-(2a_1+9d)\cdot5=735\n\n2a_1+19d-(2a_1+9d)=147\n\n10d=147\n\nd=14.7

and:  

2a_1+9\cdot14.7=47\n\n2a_1+132.3=47\n\n2a_1=-85.3\n\na_1=-42,65

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"What value does the 5 have in 893.75?A.1 / 5

B.5 / 10

C.5 / 100

D.5 / 1000"

Answers

I agree with the person above - the correct answer is C. 5 / 100
This is because 5 / 100 = 0.05 - so you see there are two digits after the comma
C. 5 / 100 hope this helps!

3 2/3 = 2 + N/3 solve for N

Answers

If the question is 3*2/3=2+N/3
N=0

If the question is 2/3=2+N/3
N=-4

Linda walked 3/4 of the length of the Tremont Trail before stopping for a rest. How far had Linda walked on the trail? Tremont Trail:3 1/2 miles Seton Trail:1 1/4 miles Wildflower Trail:2 3/8 miles 15 POINTS WILL MARK BRAINLIEST PLZ HURRY

Answers

Answer:

2 5/8 miles or 2.625 miles

Step-by-step explanation:

We are told in the question that

Linda walked 3/4 of the length of the Tremont Trail before stopping for a rest.

From the question, the length of the Tremont Trail = 3 1/2 miles

Hence, the distance (how far )Linda has walked =

3/4 of 3 1/2 miles

= 3/4 × 3 1/2 miles

= 3/4 × 7/2 miles

= 21/8 miles

= 2 5/8 miles or 2.625 miles

Therefore, Linda had walked 2 5/8 miles or 2.625 miles on the Tremont trail

Answer: the awnser is 2 5/8

Step-by-step explanation:

A coffee machine can be adjusted to deliver any fixed number of ounces of coffee. if the machine has a standard deviation in delivery equal to 0.4 ounce, what should be the mean setting so that an 8-ounce cup will overflow only 0.5% of the time?

Answers

The mean will be calculates as follows; from the formula of z-score we shall have:
z=(x-μ)/σ
the z-score associated with 0.5% is 0
thus
0=
(8-μ)/0.4=0
solving for x we get:
μ=9 

Answer: μ=9

Final answer:

The question can be addressed using the principles of Normal Distribution. Given the z-chart, 8 ounces is the observed value for the 99.5th percentile, which equates to approximately 2.58 standard deviations. Therefore, the mean setting of the coffee machine should be set around 8 ounces for the cup to overflow only 0.5% of the time.

Explanation:

The situation described in the question is a typical case of application of Normal Distribution. As a reminder, in a Normal Distribution, 99.7% of the values lie within 3 standard deviations of the mean. The question states that the cup should overflow only 0.5% of the time. Therefore, we need to consider the 99.5% of the left side under the normal curve (as we're considering the upper limit), which corresponds to around 2.58 standard deviations under the normal curve.

Given that the standard deviation (σ) is 0.4 ounces, using the formula X = μ + Zσ (where Z is the Z-score corresponding to the desired percentile, μ is the mean we want to find, and X is the threshold value where the cup overflows at 8 ounces), we can substitute the known values and solve for μ.

Therefore, 8 = μ + 2.58 * 0.4 Solving for μ gives us around μ = 7.966, or about 8 ounces. Hence, the mean setting of the coffee machine should be set around 8 ounces to ensure that the cup will overflow only 0.5% of the time.

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(4,?) is on the line 8x – 3y = -10. Find the other half of the coordinate

Answers

(4, ?) is on 8x - 3y = -10

The ? is the missing y coordinate.  (4, y) satisfies 8x - 3y = -10

8x - 3y = -10.      From point (4, y),  we would replace x = 4 in the equation below. y remains the unknown.

8*4 - 3y = -10

32 - 3y = -10

-3y = -10 - 32

-3y = - 42

y = -42/-3

y = 14
We need to plug 4 in place of x into the equation of the line and find the y-value.

8x - 3y = -10
8(4) - 3y = -10
32 - 3y = -10

Add 3y on both sides.

32 = -10 + 3y

Add 10 on both sides.

42 = 3y

Divide 3 on both sides.

y = 14

The point is (4, 14).

Translate the verbal phrase into an inequality. All real numbers that are greater than or equal to -1.5 and less than 9.2

Answers

Answer: -1.5 ≤ x < 9.2

Step-by-step explanation:

Hi, to answer this question we have to analyze the information given:

"All real numbers": we will use the variable x to represent all the real numbers.

"All real numbers that are greater than or equal to -1.5". This means that x is greater or equal (≤) to the number 1.5.

"And less than 9.2": less is represented with the symbol <  

Mathematically speaking:

-1.5 ≤ x < 9.2

Feel free to ask for more if needed or if you did not understand something.


4 is because it is greater and less than
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