Question

Write an explicit rule and recursive rule for each sequence. 1 2 3 4 5 1 3 5 7 9

Answer 1

The given sequence is

$1,2,3,4,5$

Explicit Rule:

The standard explicit formula for an arithmetic sequence is given by

$a_n=a_1+(n-1)\cdot d$

Where aₙ is the nth term, a₁ is the first term and d is the common difference

The common difference is basically the difference between any two consecutive terms

d = 5 - 4 = 1

d = 4 - 3 = 1

d = 3 - 2 = 1

So the common difference = 1

As you can see, the first term = 1

So the explicit formula for the given arithmetic sequence becomes

$a_n=1+(n-1)\cdot1_{}$

You can find any term by using the above formula.

For example:

if you want to find the 10th term then substitute n = 10 in the above formula.

Recursive Rule:

The standard recursive formula for an arithmetic sequence is given by

$a_n=a_{n-1}+d$

Where aₙ is the nth term, a(n-1) is the previous term and d is the common difference

We already know common difference = 1

So the recursive formula for the given arithmetic sequence becomes

$a_n=a_{n-1}+1$

This simply means that if you know the previous term then you add 1 to get the next term.