Use Polya's four-step problem-solving strategy and the problem-solving procedures presented in this lesson to solve the following exercise. Find the following sums without using a calculator or a formula. Hint: Apply the procedure used by Gauss. (See the Math Matters on page 31.) (a) 1 + 2 + 3 + 4 + . . . + 397 + 398 + 399 + 400 (b) 1 + 2 + 3 + 4 + . . . + 542 + 543 + 544 + 545 (c) 2 + 4 + 6 + 8 + . . . + 72 + 74 + 76 + 78
Answers
Final answer:
This answer uses Gauss's method to effectively find the sum of arithmetic series without a calculator or formula. By pairing numbers at the start and end of the series, a constant sum is found which can be easily multiplied by the number of pairs, providing the total sum of series. The sums are 80200, 149238, and 1560 respectively for the given scenarios.
Explanation:
To solve a series of summations without the use of a calculator or formula, we can apply a method used by Gauss. This problem relates to the concept of arithmetic series in mathematics.
- Step 1: Understand the problem. We're asked to find the sum of certain series. This involves adding all the numbers in the series together.
- Step 2: Devise a plan. The strategy that Gauss used was to pair numbers at the start and end of an arithmetic series. This way, each pair totals the same sum, making it easier to solve.
- Step 3: Carry out the plan.
- (a) Sum of 1 to 400: Pair numbers (1+400, 2+399, 3+398, etc.), every pair equals 401. There're 200 pairs, so 401*200 = 80200.
- (b) Sum of 1 to 545: Same strategy. Each pair (1 + 545, 2 + 544, 3 + 543, etc.) equals 546. There're 273 pairs, so the sum is 546*273 = 149238.
- (c) Sum of even numbers from 2 to 78: Notice it is an arithmetic series (2, 4, ...78). The first term a = 2 and last term l = 78. There're 39 terms. By using formula for arithmetic series (n/2)*(a + l), we have (39/2) * (2 + 78) = 1560.
- Step 4: Look back. Confirm the plan was executed correctly by reviewing the steps taken.
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Related Questions
Help please............
In an effort to make children’s toys safer and more tamperproof, toy packaging has become cumbersome for parents to remove in many cases. Accordingly, the director of marketing at Toys4Tots, a large toy manufacturer, wants to evaluate the effectiveness of a new packaging design that engineers claim will reduce customer complaints by more than 10 percentage points. Customer satisfaction surveys were sent to 220 parents who registered toys packaged under the old design and 220 parents who registered toys packaged under the new design. Of these, 81 parents expressed dissatisfaction with packaging of the old design, and 41 parents expressed dissatisfaction with packaging of the new design.What's the T-Statistic
You buy six pens for $2.99 each, and sales tax is 10%. How much change should you receive from a clerk if you give her a $20 bill?
The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives 16 years; thestandard deviation is 1.7 years.Use the empirical rule (68 - 95 - 99.7%) to estimate the probability of a gorilla living longer than 14.3years.Percent % pls
B (1,1),(3,9), (7,49)
© (1,2), (4,16), (8, 64)
D (4,8), (5, 25), (6,36)
Answers
Answer:
B (1, 1),(3, 9), (7, 49)
Step-by-step explanation:
Given function:
- y = x²
Let's verify which set of pairs are same with the given function:
A....................
- (1, 1) - yes
- (2, 4) - yes
- (3, 6) - no, 6≠ 3²
B....................
- (1, 1) - yes
- (3, 9) - yes
- (7, 49) - yes
C....................
- (1, 2)- no, 2≠ 1²
- (4, 16) - yes
- (8, 64) - yes
D....................
- (4, 8) - no, 8 ≠ 4²
- (5, 25) - yes
- (6, 36) - yes
Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
What does it mean to say that polynomials form a system analogous to integers, as
related to closure?
Answers
Answers
Answer:
The age of Jenny=21 years
The age of Steve=9 years
Step-by-step explanation:
So... Let's make their age into unknown variables so it's easier to decipher:
Jenny's age: x
Steve's age: y
If 5 years ago Jenny was four times older than Steve the equation would look like:
y+12-4y=-15
First, you add like terms:
y+12-4y=-15
y-4y=-15-12
Now isolate the unknown variable:
3y=-27
y=9
But this was 5 years ago, and Jenny is 12 years older than Steve so add those two together:
9+12=21
Hope this helped!
What is the area of the trapezoid?
Answers
Answer:
Area of trapezoid is:
162 cm²
Step-by-step explanation:
Area(A) of trapezoid is given by:
Here, we are given Height=9 cm
One parallel side=6 cm
Other parallel side=(12+6+12) cm
= 30 cm
Sum of parallel side=(6+30) cm
= 36 cm
Area=
= 162 cm²
Hence, Area of trapezoid is:
162 cm²
A = 1/2(36)(9)
A = 162
answer
162 cm^2
Answers
Answer:
a. Covariance between x and y = – 1.25
b. Correlation coefficient = – 0.07
Step-by-step explanation:
Note: This question is not complete. The complete question is therefore provided before answering the question as follows:
Consider the following sample data:
x 10 7 20 15 18
y 22 15 19 14 15
Required:
a. Calculate the covariance between the variables. (Negative value should be indicated by a minus sign. Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal place.
b. Calculate the correlation coefficient (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal place.)
The explanation to the answer is now given as follows:
Note: See the attached excel file for the calculations of the sum of x and y, means of x and y, deviations of x and y, multiplications of deviations of x and y, and others.
a. Calculate the covariance between the variables. (Negative value should be indicated by a minus sign. Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal place.)
In the attached excel file, we have:
N = Number of observations = 5
Mean of x = Sum of x / N = 70 / 5 = 14
Mean of y = Sum of y / N = 85 / 5 = 17
x - Mean of x = Deviations of x = see the attached excel file for the answer of each observation
y - Mean of y = Deviations of y = see the attached excel file for the answer of each observation
Multiplications of the deviations of x and y = (x - Mean of x) * (y - Mean of y) = see the attached excel file for the answer of each observation
Sum of the multiplications of deviations of x and y = Sum of ((x - Mean of x) * (y - Mean of y)) = –5
Since we are using a sample, we use (N – 1) in our covariance between x and y as follows:
Covariance between x and y = Sum of ((x - Mean of x) * (y - Mean of y)) / (N – 1) = –5 / (5 – 1) = –5 / 4 = –1.25
b. Calculate the correlation coefficient (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal place.)
The correlation coefficient can be calculated using the following formula:
Correlation coefficient = Covariance between x and y / (Sum of (x - Mean of x)^2 * Sum of (y - Mean of y)^2)^0.5 ………………… (1)
Where, from the attached excel file;
Covariance between x and y = –5
Sum of (x - Mean of x)^2 = 118
Sum of (y - Mean of y)^2 = 46
Substituting the values into equation (1), we have:
Correlation coefficient = –5 / (118 * 46)^0.5 = –5 / 5,428^0.5 = –5 / 73.6750 = – 0.07
Final answer:
The covariance between two variables can be calculated by first finding the mean of each dataset, subtracting the mean from each data point, multiplying the results for each pair of coordinates, summing these products to obtain the numerator. The denominator is obtained by subtracting one from the number of data points. The covariance is then the numerator divided by the denominator.
Explanation:
The term covariance is one of the key factors for understanding correlation between two variables. To calculate the covariance between the two given variables, we first need to calculate the mean of each set (x and y). After we've gotten the mean, we subtract the mean from each data point and multiply the results for each pair of x and y values. Summing these products will give us the numerator in the covariance calculation. The denominator is calculated by subtracting one from the total number of data points we have (n-1). So, the covariance is the sum we got from the numerator, divided by the denominator. Please don't forget to indicate if the covariance is negative, using a minus sign.
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Answers
Answer: Set up the ratio as a fraction and divide by the gallons. 90/5 = 18/1 The ratio is 18 miles/gallon.