Exploring the Relationship Between the First Term and the Common Difference in an Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, d, to the preceding term. The first term is denoted as a. This article delves into the relationship between the first term a and the common difference d when a certain condition is met in the sequence.
Understanding the Given Condition
The given condition states that ten times the 10th term of an arithmetic sequence is equal to fifteen times the 15th term. Mathematically, we express this as:
10 times; a10 15 times; a15
10(a 9d) 15(a 14d)
10a 90d 15a 210d
15a - 10a 90d - 210d
5a -120d
a -24d
From the above derivations, we see that there is a direct relationship between the first term a and the common difference d. Specifically, the first term is proportional to the negative of the common difference, with a constant of proportionality of -24. This relationship is also demonstrated through a simple linear sequence where these values are integers.
General Formula for the nth Term
The general term of an arithmetic sequence can be expressed as:
an a (n - 1)d
For n 1, the first term a is simply a. For n 2, the second term is a d, and so on.
Given the relationship a -24d, we can substitute d -a/24 into the general term formula to find the value of the terms in the sequence.
Example Sequence
Let's take a specific example to illustrate this relationship more concretely. Consider an arithmetic sequence starting with d -1. Given the relationship a -24d, we find:
If d -1, then a 24.
Thus, the sequence starts with 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10.
The sequence will continue with negative terms as the common difference is -1, and the first term is 24.
Verification Through Programming
A simple program can be written to verify the derived relationship for a range of values. Here is a simple C program using Microsoft QuickC version 2.00 to read values of a and d and print out the pairs that satisfy the given condition.
include stdio.hint main(void) { register int a -64; // range -64 to 65 register int d -64; // range -64 to 65 signed int term_10, term_15; for (a -64; a
The output of this program confirms several pairs of a and d that satisfy the given condition.
Discussion and Conclusion
The relationship derived, a -24d, is significant as it provides a clear and direct link between the first term and the common difference in an arithmetic sequence. This relationship is consistent with the derived formula and can be used to generate sequences or to solve for specific terms given one of the variables.
By understanding and applying this relationship, we can better analyze and manipulate arithmetic sequences, making it a valuable tool in various mathematical and real-world applications.