Question

The average of first 99 even numbers is : (a) 9999 (b) 100 (c) 9801 (d) 9009

Short answer

Answer: (b) 100

Step-by-step solution

1. Calculate the Sum of First 99 Even Numbers

To calculate the sum of the first 99 even numbers, we will use the arithmetic sequence formula. The formula for the sum of an arithmetic sequence is: Sum ($S_n$) = $\frac{n}{2}(a_1+a_n)$ Where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. Since we are working with even numbers, the first term ($a_1$) is 2, and the number of terms is 99. To find the last term ($a_n$), we will use the formula for the nth term of an arithmetic sequence: $a_n=a_1+(n-1)\ast d$ Where $d$ is the common difference between terms, which is 2 for even numbers. Therefore, $a_{99}=2+(99-1)\ast 2=2+98\ast 2=198$ Now we can use the sum formula: $S_{99}=\frac{99}{2}(2+198)=\frac{99}{2}\ast 200$

2. Calculate the Average

To find the average, divide the sum by the number of terms: Average = $\frac{S_{99}}{99}=\frac{\frac{99}{2}\ast 200}{99}$ Simplify the equation by canceling out the 99: Average = $\frac{200}{2}=100$

3. Choose the Correct Answer

Now we can identify the correct choice from the given options: The average of the first 99 even numbers is 100. Therefore, the correct answer is (b) 100.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same.
This difference is called the common difference.For example, consider the sequence of even numbers: 2, 4, 6, 8, and so on.
Here, the common difference is 2, since each number is 2 more than the preceding one.
In general, you can find the nth term of an arithmetic sequence using the formula:

• $a_n=a_1+(n-1)\cdot d$

where $a_1$ is the first term, $n$ is the term you're looking for, and $d$ is the common difference.
This formula helps in determining any term in the sequence easily.

Sum of Even Numbers

To find the sum of the first few even numbers, you utilize the arithmetic sequence's sum formula.
The sum formula will help you add up all numbers in the sequence efficiently, without needing to write them all out. The formula for the sum of an arithmetic sequence is:

• $S_n=\frac{n}{2}(a_1+a_n)$

Here, $n$ is the number of terms you intend to sum.
$a_1$ and $a_n$ are the first and last terms of the sequence.
For the first 99 even numbers:

• First term, $a_1=2$
• Last term, $a_{99}=198$

Now plug them into the formula to get the total sum.

Arithmetical Progression

Arithmetical progression is another name for an arithmetic sequence.
It emphasizes the incremental nature of the progression, which is by a constant difference each time. Understanding the arithmetic progression is crucial because it enables problem-solving with predictable, evenly-spaced numbers.
Concepts like these are utilized in real-world scenarios such as finance, predicting loan payments, or any situation where growth or reduction happens at a constant rate over time. In many math problems, identifying a sequence as an arithmetic progression allows you to use specific formulas to find unknown terms, sum terms efficiently, or calculate averages.

Average Calculation

Calculating the average of numbers in a sequence involves summing the numbers and then dividing by the count of terms.The formula for calculating an average is:

• Average = $\frac{\text{Sum of Terms}}{\text{Number of Terms}}$

For the average of the first 99 even numbers, since the sum was found earlier to be $\frac{99}{2}\times 200$, the division simplifies to:

• Average = $\frac{200}{2}=100$

This simple calculation confirms the average easily. Knowing how to calculate averages is vital in many statistical and real-life scenarios, helping us understand central tendencies of data.