A sequence is a list of numbers that follow a specific pattern. Each number in the sequence is called a term. Sequences can be finite (meaning they have a specific number of terms) or infinite (meaning they continue without end).
Arithmetic sequences are a type of sequence where each term is found by adding a constant value to the previous term. This constant value is called the common difference, denoted as d. The formula for the nth term (or any term) of an arithmetic sequence is:
an = a1 + (n-1)d
- an is the nth term of the sequence
- a1 is the first term of the sequence
- n is the position of the term we want to find (e.g. n=5 means we want to find the 5th term)
- d is the common difference between terms
In other words, we can find any term of an arithmetic sequence by adding the common difference to the previous term.
(i) The next term in the sequence is 27.
(ii) To get this answer, we can see that each term is 6 greater than the previous term
b) To find the tenth term we can write the formula for the nth term of the sequence using the common difference d = 6 and the first term a1 = 3:
an = a1 + (n - 1)d
Substituting n = 10, a1 = 3, and d = 6, we get:
a10 = 3 + (10 - 1)6
a10 = 3 + 54
a10 = 57
Therefore, the 10th term of the sequence is 57