Introduction to Finding the First (n) Terms of a Sequence Using Summation Formula

When dealing with sequences in mathematics, particularly in the context of an arithmetic progression (AP), one common task is to find the first (n) terms given the sum of the first (n) terms. This article will guide you through a detailed process of finding the first three terms of a sequence where the sum of the first (n) terms is given as (S_n n^2 - 3n).

Step-by-Step Explanation

To find the first three terms of the sequence, we need to use the relationship between the sum of terms and the individual terms. The formula to find the (n)-th term of the sequence, denoted as (a_n), is:

(a_n S_n - S_{n-1})

Calculating (S_1), (S_2), and (S_3)

First, let's calculate (S_1), (S_2), and (S_3):

(S_1 1^2 - 3 cdot 1 1 - 3 -2 20 20) (S_2 2^2 - 3 cdot 2 4 - 6 -2 34 34) (S_3 3^2 - 3 cdot 3 9 - 9 -2 42 42)

Calculating the First Three Terms (a_1), (a_2), and (a_3)

Now, we can find the first three terms using the formula:

(a_1 S_1 20) (a_2 S_2 - S_1 34 - 20 14) (a_3 S_3 - S_2 42 - 34 8)

Thus, the first three terms of the sequence are:

(a_1 20) (a_2 14) (a_3 8)

More General Case: Finding the (n)-th Term

For a more general case, we will derive the formula to find the (n)-th term of the sequence. The (n)-th term can be found by:

(a_n S_n - S_{n-1} n^2 - 3n - (n-1)^2 3(n-1))

Expanding and simplifying:

(a_n n^2 - 3n - (n^2 - 2n 1 3n - 3) n^2 - 3n - n^2 2n - 1 - 3n 3) (a_n -6n 3)

Therefore, the sequence in question is given by:

(20, 14, 8, 2, ldots)

This is an arithmetic progression (AP) with a common difference of (-6).

Conclusion

By detailing the process of finding the first three terms and the general (n)-th term of the sequence, we have shown that the sequence is an arithmetic progression with a common difference of (-6). This approach can be applied to similar problems where the sum of terms is given, allowing for the identification and calculation of individual terms of a sequence.