Arithmetic Sequences

Explicit Definition of a Sequence

An arithmetic sequence is a sequence of numbers whose difference of any two consecutive terms is constant. This difference is called the common difference.

Example: 4,7,10,13,.....
The common difference is 3 and is found by subtracting any term from the one after it.

To find the nth term, use the formula t_n = t₁ + (n - 1)d where t₁ is the first term, n is the number of the term being found, and d is the common difference.

Problem: Find a formula for the sequence 4,7,10,13,... and sketch the graph of the sequence. Then find the slope of the line containing the sequence.

Answer: t_n = 4 + (n - 1)(3)
= 4 + 3n - 3
= 3n + 1

              To sketch the graph, plot some points such as (1,4), (2,7), (3,10),... where x is the
              number of the term being graphed and y is its value. The graph is a set of ordered pairs,
              not a line. The points are linear in nature, but they are not connected.

              The slope of the line containing the sequence is found the same way the slope of any line
              is found. Select two points and find     y₂ - y_1.
                                                      x₂ - x₁

Finding the Sum of a Sequence, called a Series

Find the sum of the sequence 4,7,10,13,16,19,22,25,28,31,34,37,40.

S = 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 + 34 + 37 + 40

Also, S = 40 + 37 + 34 + 31 + 28 + 25 + 22 + 19 + 16 + 13 + 10 + 7 + 4

Now add the two sequences together:

2S = 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44 + 44

2S = 13(44)

2S = 572

S = 286

This is the same as using the formula S_n = n (t₁ + t_n) or S_n = n [2t₁ + (n - 1)d]
2 2
The second formula is obtained by substituting the formula for t_ninto the first one.

Now, substituting n = 13, t₁ = 4, and d = 3, S will turn out to be 286.

Recursive Definition of a Sequence

A recursive definition consists of two parts:

an initial condition that tells where the sequence starts

a recursion formula that tells how any term in the sequence is related to the preceding term.

For the above sequence 4, 7, 10, 13, ..., the recursive formula is

t₁ = 4
t_n = t_n-1 + 3

Problems

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