# Average of Number Series

Average of Number Series is an important concept in Quantitative Aptitude. A series is a sequence of certain consecutive numbers or terms. A sequence may consist of evenly spaced numbers. As is the case in arithmetic progressions. In other cases, the number may have progressively increasing spaces as is the case with the geometrical progressions. Here in the following space, we will see how to conveniently find the Average of Number Series. Let us begin!

## Average of Number Series

Ley us see these with the help of a few examples. A number series with a few terms may be written down and the average will be found using the usual formula. The formula of average = (Sum of observations)/(Number of observations).

What about the series that have hundreds of terms? There are many methods that we can use to find the average of such series, here we shall see most of them. Let us start with the first one.

### Method I – Averaging Any Short Series of Consecutive Numbers

The first step is that we count the number of terms in the series. This is the number of terms or numbers in the sequence. Determine whether the series has an odd or even number of terms. For example, the sequence 3, 4, 5, 6, 7, 8, 9 has seven terms, an odd amount. The sequence 3, 4, 5, 6, 7, 8 has six terms, an even amount.

The second step is to identify the middle number of a series with an odd number of terms. This is the number that has the same amount of terms on either side of it. This middle number will be the average of the series. For example, in the sequence 3, 4, 5, 6, 7, 8, 9, the middle number is 6. It has three numbers to the left of it, and three numbers to the right of it. So, in this series of numbers, 6 is both the mean and the median.

### Average the middle numbers of a series with an even number of terms

To do this, find the pair of numbers that have the same amount of terms on either side of it. To find the average, add these two numbers together and divide by two. Their average will be the average of the series. For example, in the sequence 3, 4, 5, 6, 7, 8, the middle pair is 5 and 6. It has two numbers to the left of it, and two numbers to the right of it. So, to calculate the average of the series, calculate the average of these two numbers:
(5+6)/2 = 11/2 = 5.5
So, in this series of numbers, 5.5 is both the mean and the median.

### Method II – Averaging Any Long Series of Consecutive Numbers

The evenly spaced set of numbers. Let us say that we have an A. P. which has several evenly spaced numbers. Let the first term of the sequence be x1 and the last term be xn. Then the average of the series can be found out by the following formula:

Average = (x1 + xn)/2.

For example, if you were finding the average of sequential numbers beginning with 15 and ending with 45, your formula will look like this: Average = (15 + 45)/2 = 30. So, the average of the series of consecutive numbers beginning with 15 and ending with 45 is 30.

### Method III – Averaging Any Consecutive Series Beginning with 1

Let S represent the sum of all the numbers in such a series, and n equal the number of terms in the series. Then, since the series begins with one, then the number of the terms will be equal to the last term in the series. We can find the sum of such a series by the following formula:

S = n(n + 1)/2; Thus we can find the sum of such a series. The average is still given by the formula, S/n; where n is the number of terms. Here the important step is to find the number of terms n. Once you have found the number of terms, you can easily find the sum of the numbers.

For example, let us see a few number series and try and find their average.

Example 1: Find the average of the first hundred numbers?

A) 50                    B) 49                    C) 49.5                     D) 50.5

Answer: You may think that the answer is A) 50 but that is not correct. Let us write the series below:

1, 2, 3, 4, 5, …, 49, 50, 51, …, 98, 99, 100. There is a total of hundred terms in the series which begins with 1. Therefore the third method shall be applicable here. First of all, we will find the sum of the series and then divide this sum by the number of terms to get the average.

We know that sum, S = n(n+1)/2. In other words, we may write, S = 100(100 + 1)/2 = 50(101) = 5050. Therefoe the average will be = 5050/100 = 50.5 and hence the correct answer is D) 50.5.

Here we can cut a step and write the formula for the average of such a series i.e. one that begins with 1 and has consecutive numbers, as Average = n(n+1)/2n or Average = (n+1)/2. Now let us solve the following questions to get a better grip on the topic.

## Average Speed And Confusions

There are a few average speed formulae that you need to know before we move ahead. They are very important from an exam point of view.

1. Average Speed = (a + b)/2

Applicable when one travels at speed a for half the time and speed b for another half of the time. In this case, the average speed is the arithmetic mean of the two speeds.

2. Average Speed = 2ab/(a + b)

Applicable when one travels at speed a for half the distance and speed b for another half of the distance. In this case, the average speed is the harmonic mean of the two speeds. On similar lines, you can modify this formula for one-third distance.

3. Average Speed = 3abc/(ab + bc + ca)

Applicable when one travels at speed a for one-third of the distance, at speed b for another one-third of the distance and speed c for rest of the one-third of the distance. Note that the generic Harmonic mean formula for n numbers is

Harmonic Mean = n/(1/a + 1/b + 1/c + …)

4. You can also use weighted averages. Note that in case of average speed, the weight is always ‘time’. So in case you are given the average speed, you can find the ratio of time as:

t1/t2 = (a – Avg)/(Avg – b). As you already know, this is just our weighted average formula. Now, let’s look at some simple questions where you can use these formulas.

## Practice Question:

Q 1: A car is travelling at a constant velocity of 5 m/s. At a certain point in its journey, the car starts to accelerate at regular intervals such that its speed gets to 10 m/s. If the distance travelled is equal in both the cases, then the average velocity of the car after ten seconds is:

A) 6.67 m/s                 B) 8.67 m/s                  C) 12.77 m/s                      D) 14 m/s

Ans: A) 6.67 m/s

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