Calculating the Arithmetic Mean of an AP

The question asks for the arithmetic mean of the first 50 terms of an Arithmetic Progression (AP) where the first term is 4 and the common difference is 4.

Understanding Arithmetic Mean of an AP

The arithmetic mean of a set of numbers is their sum divided by the count of numbers. For an Arithmetic Progression, there are a couple of ways to find the arithmetic mean:

• The arithmetic mean of an AP is equal to the average of its first and last term.
• Alternatively, it is the sum of all terms divided by the number of terms.

Given Information

• First term ($a_1$ or $a$): 4
• Common difference ($d$): 4
• Number of terms ($n$): 50

Step-by-Step Calculation

Step 1: Find the last term (50th term) of the AP

The formula for the n-th term of an AP is $a_n = a + (n-1)d$.

Here, $n = 50$, $a = 4$, and $d = 4$.

So, the 50th term ($a_{50}$) is:

$$a_{50} = 4 + (50-1) \times 4$$ $$a_{50} = 4 + 49 \times 4$$ $$a_{50} = 4 + 196$$ $$a_{50} = 200$$

The last term (50th term) of the AP is 200.

Step 2: Calculate the Arithmetic Mean

Using the property that the arithmetic mean of an AP is the average of the first and last term:

$$\text{Arithmetic Mean} = \frac{\text{First Term} + \text{Last Term}}{2}$$ $$\text{Arithmetic Mean} = \frac{a_1 + a_{50}}{2}$$ $$\text{Arithmetic Mean} = \frac{4 + 200}{2}$$ $$\text{Arithmetic Mean} = \frac{204}{2}$$ $$\text{Arithmetic Mean} = 102$$

The arithmetic mean of the 50 terms of the AP is 102.

Alternative Calculation Method (Using Sum of AP)

Step 1: Find the sum of the 50 terms of the AP

The formula for the sum of the first n terms of an AP is $S_n = \frac{n}{2}(a + a_n)$.

Using $n=50$, $a=4$, and $a_{50}=200$:

$$S_{50} = \frac{50}{2}(4 + 200)$$ $$S_{50} = 25(204)$$ $$S_{50} = 5100$$

The sum of the first 50 terms is 5100.

Step 2: Calculate the Arithmetic Mean

The arithmetic mean is the sum of the terms divided by the number of terms.

$$\text{Arithmetic Mean} = \frac{S_n}{n}$$ $$\text{Arithmetic Mean} = \frac{S_{50}}{50}$$ $$\text{Arithmetic Mean} = \frac{5100}{50}$$ $$\text{Arithmetic Mean} = 102$$

Both methods yield the same result, confirming the arithmetic mean is 102.

Summary of Calculations

Final Answer

The arithmetic mean of the 50 terms of the AP is 102.

Revision Table: Key AP Concepts

Additional Information: Properties of Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference ($d$).

• If $d > 0$, the AP is increasing.
• If $d < 0$, the AP is decreasing.
• If $d = 0$, the AP is constant.

The terms of an AP can be represented as $a, a+d, a+2d, a+3d, \dots$.

A key property used in finding the mean is that terms equidistant from the beginning and end have the same sum. For example, $a_1 + a_n = a_2 + a_{n-1} = a_3 + a_{n-2}$, and so on. The arithmetic mean is effectively the value of the middle term if the number of terms is odd, or the average of the two middle terms if the number of terms is even. In our case, with 50 terms, the mean is the average of the 25th and 26th terms, which is equal to the average of the 1st and 50th terms.