n-th Terms of Linear Sequences

Understanding Linear Sequences

Linear sequences can also be called arithmetic sequences. They follow a simple pattern of adding or subtracting a consistent value.

A linear sequence has a constant difference between terms - this is often referred to as the common difference
The common difference can be found by subtracting any term from the term that follows it.
The n-th term of a linear sequence: This is a formula that allows you to find any term in the sequence without listing all the preceding terms.

Identifying the n-th Term

The n-th term of a linear sequence can be found using the formula a + (n - 1)d, where “a” is the first term in the sequence, “d” is the common difference, and “n” is the term number in the sequence.
Example: For the linear sequence 5, 7, 9, 11, …, a = 5 and d = 2.The n-th term is therefore 5 + (n - 1)2. Simplifying, we get 2n + 3.

Application of the n-th Term

Once you have found the n-th term, you can find any term in the sequence.
Example: If the n-th term formula is 3n + 2, the 10th term of the sequence is 3*10 + 2 = 32.

Remember that understanding the concept of n-th term is useful not just for finding terms in a sequence, but also for determining whether a number is part of a sequence.