Average and Arithmetic Progression – CAT 2022

Wordpandit

3 years ago

In this article, we will discuss the method to calculate averages. We will also discuss about a special series i.e.,the Arithmetic Progression (AP) and ways to calculate averages in an AP.

Essentially, an Average is the estimation of the middle number of any series of numbers.
For example, the average of 1,2,3,4,5 is 3. We can simply say that:

The Average of 1,2,3,4,5 =

This number is also the middle term of the series. Using this example, we can say that in an A.P. (i.e., arithmetic progression), the middle term is the average of the series.

Note: The middle term is the average of the series when the terms are arranged in ascending or descending order.

But this does leave us with one small issue. What if the A.P. has an even number of terms? Which is the middle term in this case?

In the Arithmetic Progression, there are two cases, first, when the number of terms is odd and the second one is when the number of terms is even.
1. When the number of terms is odd, the average will be the middle term.
2. When the number of terms is even, the average will be the average of two middle terms.

For the terms in A.P., we can also use (first term + last term)to find the average of the given terms.

Example: What will be the average of 13, 14, 15, 16, 17?
Solution: Average is the middle term when the number of terms in an A.P. is odd. Therefore, let us check whether the given series is an A.P. or not. Since the common difference of the terms is same, therefore, the series is in A.P.
So, the middle term 15 is our average of the series.
Let us check it in another way. We know that the average of a set of terms is equal to:
Average =
So, the sum of all terms in this case is 75 and the number of terms is 5 so the average is 15.
Yet another way would be to use the formula for terms in A.P.,
Average = (first term + last term)=

Now let us come to the second form when the number of terms is even.
Example: What will be the average of 13, 14, 15, 16, 17, 18?
Solution: We have discussed that when the number of terms is even, then the average will be the average of two middle terms.
Now, the two middle terms are 15 and 16, but before calculating the average we must check whether the series is an A.P. or not.
Since the common difference between the adjacent terms is same, we can say that the series is in A.P. and thus, the average is (16+15)/2 = 15.5
Another way is by using average’s basic concept,
Average =

Yet another way is to use formula for terms in A.P.,
average = (first term + last term) =

Now, let us put the above discussed concepts to use by solving the questions on averages in the given exercise.

EXERCISE

Question 1: The average of 7 consecutive numbers is 20. The largest of these numbers is:
(1) 24
(2) 23
(3) 22
(4) 20
(SSC Combined Graduate Level Prelim Exam. 27.02.2000 (First Sitting)

Answer and Explanation

Solution: Option 2

Average of 7 consecutive numbers is 20. Since the numbers are consecutive, they form an arithmetic series with common difference 1. We know when the number of terms is odd the average will be the middle term. So middle term is 20. Hence there will be 3 numbers before 20 and 3 numbers after 20. So complete series can be written as 17, 18, 19, 20, 21, 22, 23 Therefore, the largest of these numbers is 23.

Question 2: The average of first nine prime numbers is

(1) 9

(2) 11

(3) 11

(4) 11

(SSC CPO Sub-Inspector Exam. 12.01.2003)

Answer and Explanation

Solution: Option 4

First nine prime numbers are 2,3,5, 7,11,13,17, 19 and 23.

These are not in arithmetic progression, so do not apply the above discussed concept of odd-even here.

Here, we will use the formula Average =

Average==

Question 3: The arithmetic mean (average) of the first 10 whole numbers is
(1) 5
(2) 4
(3) 5.5
(4) 4.5
(SSC CISF ASI Exam 29.08.2010 (Paper-1)

Answer and Explanation

Solution: Option 4

First 10 whole numbers are 0,1,2,3,4,5,6,7, 8 and 9.

When the number of terms is even then the average will be the average of two middle terms.

Or Use formula that average = (first term + last term) =

Question 4: The average of first five odd multiples of 3 is

(1) 12

(2) 16

(3) 15

(4) 21

(SSC (10+2) Level Data Entry Operator & LDC Exam. 04.11.2012 (II^nd Sitting)

Answer and Explanation

Solution: Option 3

The first 5 odd multiples of 3 are 3,9,15,21 and 27. Since the difference between the terms is same, the series is an A.P. and the since the total terms are odd, the middle term will be the average. Hence the required average is 15.

Question 5: What will be the average of first 10 natural numbers?
(1) 5.5
(2) 5
(3) 5.75
(4) 4.75

Answer and Explanation

Solution: Option 1

Since the first 10 natural numbers are 1,2,3,4,5,6,7,8,9 and 10.

So, the average = (first term + last term) =

Averages Questions: Problems on averages you should solve for competitive examination preparation

Welcome to this exercise on Problems on Averages. In this exercise, we build on the basic concepts for finding the Average. As you explore this topic, you will come across questions where you will be needing to find averages while also dealing with Arithmetic Progression. Such questions need optimized tackling and can be solved with ease by using the formulas and understanding the relationships highlighted in this Averages Questions article. The Averages Questions exercise comes into the picture where it highlights the important concepts related to Arithmetic Progressions and Averages and tricks you should keep in mind for this question type.