Understanding the Sequence Defined by an n - 1
" "The sequence defined by the formula an n - 1 is a simple yet concise way to describe a pattern in a series of numbers. Each term in the sequence is derived from the value of n, where n represents the position of the term in the sequence. This formula allows us to determine the first few terms directly, making it easy to understand and visualize the pattern.
" "Deriving the First Four Terms
" "Let's start by finding the first four terms of the sequence:
" "" "For the first term, when n 1:" "a1 1 - 1 2" "For the second term, when n 2:" "a2 2 - 1 3" "For the third term, when n 3:" "a3 3 - 1 4" "For the fourth term, when n 4:" "a4 4 - 1 5" "" "Thus, the first four terms of the sequence are 2, 3, 4, and 5. This sequence is an arithmetic progression where the first term is 2 and the common difference is 1.
" "A Detailed Look at the Sequence Formula
" "The formula an n - 1 can be rewritten in a more general form where we can see the first term and the common difference more clearly. This form is:
" "an a1 (n - 1)d
" "For the sequence an n - 1, we have:
" "" "First term (a1) 2" "Common difference (d) 1" "" "Rewriting the formula in the general form, we get:
" "an 2 (n - 1)1
" "This representation confirms that the first term of the sequence is 2, and each subsequent term is obtained by adding 1 to the previous term. Therefore, the sequence is 2, 3, 4, 5, 6, and so on.
" "Verification and Application
" "Let's verify this with a few more terms to ensure the pattern holds:
" "" "For n 5:" "a5 5 - 1 4" "Since the previous term a4 5, we add the common difference, 1:" "a5 4 1 5" "" "Similarly, for n 6:
" "" "a6 6 - 1 5" "Since the previous term a5 5, we add the common difference, 1:" "a6 5 1 6" "" "We can see that the sequence continues as 2, 3, 4, 5, 6, 7, and so on.
" "Conclusion
" "In conclusion, the sequence defined by an n - 1 is an arithmetic sequence with the first term 2 and a common difference of 1. The first four terms of the sequence are 2, 3, 4, and 5. Understanding this formula and how to apply it can be very useful in various mathematical contexts and problem-solving scenarios.