Write the mathematical expression that describes the arithmetic sequence

1 answer:

6 0

There are three types of progression: arithmetic progression, geometric progression and harmonic progression. Arithmetic progression is a sequence of numbers or variables that has a common difference. Geometric progression has a common ratio, while harmonic progression is just the reciprocal of the sequence in arithmetic progression.

Example of an arithmetic sequence is: 34, 37, 40, 43, 46. The adjacent terms have a common difference of 3. Mathematicians have already derived equations for arithmetic progression so it would be just convenient for us to predict the next numbers or missing numbers in the sequence. These equations are:

An = A1 + (n-1)d
Sn = (n/2)(A1 + An)

where
An is the nth term in the sequence
A1 is the first term in the sequence
n is the total number of terms int he sequence
d is the common difference
Sn is the sum of all the terms in the sequence

These two equations describe an arithmetic sequence.

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In a recent issue of the IEEE Spectrum, 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six

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Answer:

b) 3

c) 5

d) 4

e) 3

f) 3 days to 5 days

Step-by-step explanation:

Total number of conferences = 84

Part a) We can make a pie chart to represent the distribution of number of days. The area of each sector represents the percentage of duration compared to all events. For example, the conferences that lasted 2 days occurred for 4.7619% or approximately 5% of the time.

Part b) The first quartile

The number of days for which the conferences last as per the given data in ascending order are listed below in the second image. In order to find First Quartile, we first need to find the Median. Since, total number of quantities is 84, the median will be the average of middle two values (42nd and 43rd). These values are made bold in the second image below. So median of the data is 4. Median divides the data into halves, both of which are colored separately in the image. First Quartile is the middle value of the first half. Since, values in first are 42, the middle value will be the average of central two values (21st and 22nd). These are made bold and colored red. Thus, First Quartile of the data is 3.

Part c) The third quartile

Similar to last step, the 3rd Quartile is the middle value of second half of the data. The second half of the data is colored purple. Number of values in this half are again 42, so middle value will be the average of digits at 21st and 22nd place, which comes out to be 5. Therefore, 3rd quartile is 5.

Part d) The 65th percentile.

65th percentile means, 65% of the data values are below this point. 65% of the 84 is 54.6. This means 54.6 values should be below the 65th percentile. Thus, 65th percentile occurs at 55th position. Counting from the start, the digit at 55th position comes out to be 4. So, the 65th percentile is 4.

Part e) The 40th percentile

40th percentile means, 40% of the data values are below this point. 40% of the 84 is 33.6. This means 33.6 values should be below the 40th percentile. Thus, 40th percentile occurs at 34th position. Counting from the start, the digit at 34th position comes out to be 3. So, the 40th percentile is 3.

Part f) Middle 50% of the conferences

Since, First Quartile is 25th percentile and Third Quartile is 75th percentile, in between these two Quartiles 50% of the data is present. The difference of first quartile and third quartile is known as IQR, Inter Quartile range and is a common measure of spread in stats.

Therefore, for the given data:

The middle 50% of the conferences last from 3 days to 5 days