Askjagden's Guide to Recursive and Explicit Formulas

Askjagden's Guide to Recursive and Explicit Formulas

Sequences are important elements of algebra, and are one of the stepping stones leading to calculus. Therefore I have decided to continue my guide on mathematical patterns. Please enjoy!

What is an arithmetic sequence? An arithmetic sequence is an infinite sequence of numbers (which is a sequence that goes on forever) that have a common difference. A common difference is the distance between each consecutive term, or element of each sequence. Note that the common difference must be constant between all terms.

Let's have an example. We'll call the sequence u. The initial term is 1, and the common difference is 1. To display this sequence, let's have a few terms and then show that the sequence is infinite by showing an ellipsis: 1, 2, 3 . . .

A recursive formula shows what a term is in relation to a term preceding it. In other words, you add the common difference to the term before the term you are trying to solve for, which then results in the term you were looking for. Let's use the example above and find u_4. The preceding term (u_3) is 3, and the common difference is 1; 3 + 1 = 4. Therefore u_4 = 4. (Just in case you don't know what u_4, u_3, etc. mean, the "_" stands for subscript, and the number stands for the term's position.)

For arithmetic sequences, a certain formula can be used to find any term in the sequence. That formula is known as the explicit formula. Here it is: i is a random sequence, d is the common difference, and i_n is the term you are trying to find; i_n = i_1 + d(n-1). It always works.

Let's use the arithmetic sequence -4, -2, 0, 2 . . . Let's find i_6. i_6 = -4 + 2(6-1)

                                                                                                                       -4 + 2(5)

                                                                                                                        -4 + 10

                                                                                                                             6

Therefore i_6 = 6.

What about the recursive and explicit formulas of geometric sequences? And what is a geometric sequence? A geometric sequence is an infinite sequence with a common ratio. The common ratio is the number you need to multiply and hence get the consecutive term after a number. Like the common difference, the common ratio must be constant.

Let's make a random geometric sequence g: 1, 2, 4 . . . The recursive formula for a geometric sequence is for sequence g: g_n = g_(n-1) x d (x means "times", by the way), and the explicit formula for a geometric sequence g is g_n = g_1 x r^(n-1) (^ means "to the power of").

Let's find the recursive and explicit formulas of g_4: g_4 = 4 x 2 = 8

                                                                                  g_4 = 1 x 2^3 = 1 x 8 = 8

Thus we can prove the explicit and recursive formulas.

Have fun with recursive and explicit formulas!