Solving for the First Term and Common Difference in an Arithmetic Progression
Arithmetic progressions (AP) are sequences in which each term after the first is obtained by adding a constant difference, known as the common difference (d). Understanding how to find the first term and the common difference is crucial in dealing with AP problems. In this article, we will explore how to approach such problems systematically.
Introduction to Arithmetic Progressions
An arithmetic progression can be defined by its general formula:
a_n a_1 (n-1)d
where a_n is the nth term, a_1 is the first term, and d is the common difference. To find the first term and common difference, you may need to use the given terms in the progression.
Example 1: Given the Second and Seventh Terms of an AP
Suppose the second and seventh terms of an arithmetic progression are 8 and 23, respectively. To find the common difference and the first term, follow these steps:
Write down the general formula for the nth term:
a_n a_1 (n-1)d
Substitute the known terms:
a_7 a_1 6d 23
a_2 a_1 d 8
Form two equations and solve for d:
23 a_1 6d
8 a_1 d
Subtract the second equation from the first:
15 5d
d 3
Find the first term:
a_1 8 - d 8 - 3 5
Thus, the first term is 5, and the common difference is 3. The first seven terms are: 5, 8, 11, 14, 17, 20, 23.
Example 2: Given the Fourth and Ninth Terms of an AP
Let's take another example where the fourth and ninth terms of an arithmetic progression are 10 and 20, respectively.
Write down the general formula for the nth term:
T_n a_1 (n-1)d
Substitute the known terms:
T_4 a_1 3d 10
T_9 a_1 8d 20
Solve the equations to find d:
T_9 - T_4 5d 10
d 2
T_4 a_1 3d 10
a_1 10 - 3d 10 - 3(2) 4
The first term is 4, and the common difference is 2.
Conclusion
Understanding the arithmetic progression formula and how to manipulate it is key to solving problems involving the first term and common difference. By following the steps outlined in these examples, you can systematically solve for these values in any given arithmetic progression.
Use the formulas and methods described above when dealing with similar problems. Remember, the key is to set up accurate equations and solve them step by step.