Question

The average of 3 consecutive natural numbers (which are in increasing order) is \(k\). If two more consecutive number, just next the first set of numbers, is added, then the new average \(\begin{array}{ll}\text { becomes: } & \text { (b) } k+1\end{array}\) (a) \(k+2\) (c) \(\frac{2 k+1}{2}\) (d) \(2 k-1\)

Short answer

Answer: k+1

Step-by-step solution

Let's represent the 3 consecutive natural numbers

Since the numbers are consecutive, we can represent them as (n-1), n and (n+1), where n is a natural number.

Calculate the average of the initial 3 numbers

To find the average of 3 numbers, we sum them up and divide by 3. \((n-1)+n+(n+1)=3k \Rightarrow 3n=3k \Rightarrow n=k\)

Represent the numbers to be added and initial summed up value in terms of k

We know that n=k. So, the 3 initial consecutive numbers are: \((k-1)\), \(k\), and \((k+1)\). The next 2 consecutive numbers are \((k+2)\) and \((k+3)\). And the initial sum of the first 3 numbers is \(3k\).

Calculate the new average with all 5 numbers

We sum up all 5 numbers and divide by 5 to find the new average: \(\frac{(k-1) + k + (k+1) + (k+2) + (k+3)}{5}=\frac{5k+5}{5}=k+1\) So, the new average is \((b) \ k+1\).

Key concepts

Quantitative Aptitude

Quantitative aptitude is a crucial skill that evaluates one's ability to understand numbers, solve various numerical problems, and make sound decisions using mathematical concepts. It's a common segment found in competitive exams and often includes questions on arithmetic progression, averages, percentages, and more.

When dealing with questions involving consecutive natural numbers, understanding patterns in the series is vital. Consecutive numbers form what is known as an arithmetic progression, a sequence of numbers with a constant difference between adjacent terms. This concept is fundamental to solving exercises like the one we're discussing.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. For example, the sequence 2, 4, 6, 8 is an AP with a common difference of 2.

In the presented problem, the initial AP is the 3 consecutive natural numbers. They can be expressed algebraically as (n-1), n, and (n+1), where each term increases by 1. This understanding simplifies computations, as seen in the step by step solution.

Average Calculation

The average or mean of a set of numbers is calculated by summing all the numbers and then dividing by the count of those numbers. Averages are useful to summarize a set of data with a single value that represents the center point. In our example, the average of initial 3 consecutive numbers (n-1), n, and (n+1) is \(\frac{(n-1) + n + (n+1)}{3}\), which simplifies to \(\frac{3n}{3}\) or simply n.

Subsequently, when two more consecutive numbers (k+2) and (k+3) are added to the original set, the average is recalculated with five numbers. The average reflects the central value of the 5 numbered set, which in our case, shifts to (k+1), signifying the consistency of arithmetic progression in average calculations.