Hilary is given the pattern 2, 4, 6, 8, 10,... on her test. She has to find the 50th term of the pattern. To do so she needs to describe the pattern either with words, a recursive formula, or an explicit formula. Which would be the best? Make sure you also explain why the forms you do not choose are not good representations.

A recursive formula would be best to describe the pattern.

Step-by-step explanation: The pattern of numbers in the question clearly indicates it is an arithmetic progression, that is, every number is derived by adding a common difference to the previous number. The common difference or d, does not change throughout the sequence.

The common difference in the sequence above is 2. Upon close observation we would observe that by simply adding 2 to a number we can arrive at the next number.

However, using words to describe the pattern of the sequence would not be helpful if we have to find a number very far into the sequence, for example if we were to find the 1000th term of the sequence.

A recursive formula is preferable and would be the best option because of its simplicity in application. The recursive formula to calculate the nth term of an arithmetic progression is given as

nth = a + (n - 1)d

Where n is the term to be calculated in the sequence (in this case n equals 50), a is the first term (2 in this case) and d is the common difference (2 in this case).

The 50th term can be calculated as follows;

nth = 2 + (50 - 1)2

nth = 2 + (49)2

nth = 2 + 98

nth = 100

The calculation above shows how simple it is to calculate the nth term with a recursive formula rather than with verbal descriptions.

An explicit formula also allows you to find the value of any term in a sequence. The explicit formula designates the nth term of the sequence as an expression of n, that is, it defines the sequence as a formula in terms of n. This formula lets us find any other term without knowing other terms.