Understanding an Arithmetic Sequence with First Term -10 and Common Difference -2
In mathematics, an arithmetic sequence, also known as 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. In the context of an arithmetic sequence, the first term is crucial, as it sets the starting point for the entire sequence.
The Problem
Consider an arithmetic sequence where the first term (a_1 -10) and the common difference (d -2). The task is to find the first ten terms of the sequence.
Formula for Arithmetic Sequences
To find the terms in an arithmetic sequence, we use the formula for the nth term:
[a_n a_1 (n-1)d]
where:
(a_n) (a_1) (d) (n)Solving the Problem
Let's use the formula to find each of the first ten terms in the sequence:
(a_1 -10) [a_2 -10 (2-1)(-2) -10 (-2) -12] [a_3 -10 (3-1)(-2) -10 (-4) -14] [a_4 -10 (4-1)(-2) -10 (-6) -16] [a_5 -10 (5-1)(-2) -10 (-8) -18] [a_6 -10 (6-1)(-2) -10 (-10) -20] [a_7 -10 (7-1)(-2) -10 (-12) -22] [a_8 -10 (8-1)(-2) -10 (-14) -24] [a_9 -10 (9-1)(-2) -10 (-16) -26] [a_{10} -10 (10-1)(-2) -10 (-18) -28]The First Ten Terms of the Sequence
Therefore, the first ten terms of the sequence are:
-10 -12 -14 -16 -18 -20 -22 -24 -26 -28Conclusion
Understanding how to find the terms of an arithmetic sequence is a fundamental skill in mathematics. The ability to apply the arithmetic sequence formula enables us to determine any term in the sequence, regardless of its position, based on the first term and the common difference.
Additional Resources
For more detailed information on arithmetic sequences, you can explore the following resources:
Wikipedia: Arithmetic Progression Khan Academy: Arithmetic Sequences/p