Understanding Arithmetic Progressions and Finding the First Term

An arithmetic progression (AP) is a sequence of numbers such that the difference between any two successive members is a constant. This constant difference is denoted as the common difference, denoted by 'd'. In this article, we will explore how to find the first term of an AP given the common difference and a specific term.

Key Concepts

1. Common Difference (d): The difference between any two consecutive terms in an AP. 2. nth Term of AP: The formula to find the nth term of an AP is given by: an a (n-1)d, where 'a' is the first term of the AP.

Given Information in Problem

In the original problem, we are given:

The common difference (d) -4. The seventh term (a7) 4.

We need to determine the first term (a).

Step-by-Step Solution

We start from the formula for the nth term of an AP and substitute the given values:

Step 1: Write the formula for the nth term of an AP.

a n a ( n - 1 ) d

Step 2: Substitute n 7, a7 4, and d -4 into the formula.

4 a ( 7 - 1 ) ( -4 )

Step 3: Simplify the expression.

4 a ( 6 ) ( -4 ) 4 a - 24

Step 4: Solve for the first term (a).

4 24 a 28 a

Therefore, the first term of the AP is 28.

Alternative Approach

An alternative method involves a more intuitive step-by-step approach. Starting from the seventh term, we work backwards using the common difference. Moving from the seventh term (4), we can add 4 four times to find the first term: a 7 4 a 6 4 4 8 a 5 8 4 12 a 4 12 4 16 a 3 16 4 20 a 2 20 4 24 a 1 24 4 28

The result is the same: the first term is 28.

Conclusion

As demonstrated in both methods, the first term (a) of the AP can be found using the given common difference (d) and a specified term (a7) of the sequence. Understanding the basic principles of APs and applying the relevant formulas or intuitive deduction methods are key to solving such problems effectively.