Finding the First Term of an Arithmetic Progression with Given Common Difference and Seventh Term

In an Arithmetic Progression (AP), the nth term can be expressed as:

a_n a_1 (n - 1)d,
where: a_n - is the nth term, a_1 - is the first term, d - is the common difference, and n - is the term number.

Given:

The common difference d -4 The seventh term a_7 4

Solving for the First Term

The formula for the seventh term of an AP is given by:

a_7 a_1 (7 - 1)d

Substituting the known values:

4 a_1 (6)(-4)

This simplifies to:

4 a_1 - 24

Solving for a_1:

a_1 4 24

a_1 28

Thus, the first term a_1 is 28.

Additional Homework Help on Arithmetic Progressions

Following are similar examples to help you understand the concept of finding the first term in an arithmetic progression:

Example 1:

Given: d -4, a7 4 Let the first term a nth term a (n-1)d a7 a (7-1)(-4) 4 a - 24 a 28 first term 28

Example 2:

Nth term of AP: a_n a (n-1)d

Here, n 7.

a7 a (7-1)(-4) 4

4 a - 24

a 28

Example 3:

Given: d -4, a7 4 an an-1d Here, n 7. a7 a6d 4 a 28

Example 4:

Let a_n be the nth term of the arithmetic progression. Assuming d a_{k 1} - a_k for k in mathbb{Z} then a_7 - a_1 a_7 - a_6 a_6 - a_5 ldots a_3 - a_2 a_2 - a_1 6d which means that a_1 a_7 - 6d 4 - 6(-4) 28.

Conclusion

The process of finding the first term in an arithmetic progression involves using the known common difference and the given term. By utilizing the nth term formula, the first term can be derived through simple algebraic manipulation.