Question

The average of $7,14,21,28\dots .7715=$ (a) 7 (b) 11 (d) 66 (c) 42

Short answer

Answer: (d) 66

Step-by-step solution

Identify the Arithmetic Progression (AP)

The first term is $a_1=7$ and the common difference is $d=7$.

Find the Total Number of Terms

Let $n$ be the total number of terms in the sequence. The last term $a_n=7715$. Using the formula for an arithmetic progression, we have $a_n=a_1+(n-1)d$. Substituting the given values, we get: $7715=7+(n-1)7$. Now, solve for $n$: $7708=(n-1)7$ $n-1=1101$ $n=1102$ So, there are a total of $1102$ terms in the sequence.

Find the Sum of the Terms

To find the sum of an arithmetic progression, we use the formula: $S_n=\frac{n(a_1+a_n)}{2}$ Substituting the values, we get: $S_{1102}=\frac{1102(7+7715)}{2},$ which simplifies to: $S_{1102}=4294296$.

Calculate the Average

Divide the sum by the total number of terms to find the average: Average $=\frac{S_{1102}}{1102}=\frac{4294296}{1102}=$ 66

Choose the correct answer from the options

The average of the sequence is 66, which corresponds to option (d). Therefore, the correct answer is (d) 66.

Key concepts

Average Calculation

The concept of average is essential in understanding how values are spread out across a dataset. Here, we calculate it by adding all the values together and then dividing by the total number of values.
For a sequence in arithmetic progression (AP), such as the one provided, calculating the average is made simpler with a formula once the sum of the series is known.

• First, find the sum of all the terms in the series.
• Next, divide this sum by the number of terms.

In the example provided, the sum of the terms is found to be 4,294,296, and the total number of terms is 1,102. So, the average calculation is done by dividing the sum by the number of terms, yielding the result 66.

Series and Sequences

Series and sequences are central to understanding patterns in mathematics. An arithmetic sequence, also known as an arithmetic progression (AP), consists of numbers in a sequence where each term after the first is obtained by adding a constant called the 'common difference'.

In the sequence given, the terms are 7, 14, 21, and so on, indicating a common difference of 7.

• The first term is 7.
• The formula to find any term in the sequence is given by: $a_n=a_1+(n-1)d$.
• In our example, the lattice of terms ranges until 7715.

Understanding these patterns helps quantify how sequences grow, and this arithmetic structure makes calculations such as averages or sums much more systematic and efficient.

Sum of Terms

The sum of terms in an arithmetic progression can be quickly calculated with a formula that simplifies the process.

To find the sum of the first $n$ terms of an arithmetic sequence, use the formula:

• $S_n=\frac{n}{2}(a_1+a_n)$

This formula allows you to quickly find the total without individually adding each term.

By substituting the known values (e.g., the first term $a_1$ as 7, the nth term $a_n$ as 7715, and $n$ as 1102), we found the sum to be 4,294,296. This establishes the foundation for finding other metrics in quantitative analysis, like average.

Quantitative Aptitude

Quantitative aptitude is a critical skill set that involves solving problems by applying mathematical knowledge. It often includes arithmetic, number series, and basic algebra.
Such problems can include calculating averages or determining the properties of number sequences like arithmetic progressions.
Quantitative aptitude tests are commonplace in various competitive exams and standardised tests, helping to assess an individual's ability to handle numbers efficiently.

Mastering the ability to manage and manipulate numbers through practiced techniques in sequences and averages will significantly benefit anyone aiming to excel in mathematical challenges. It's crucial to progressively learn these skills for academic progressions and real-world applications.