FIRST ENCOUNTER WITH ARITHMETIC SEQUENCES

Consider the following sequences of real numbers.

Sequence 1:     2, 5, 8, 11, 14, …

Sequence 2:     11, 16, 21, 26, 31, …

Sequence 3:     15, 13, 11, 9, 7, …

They are examples of what we call arithmetic sequence or arithmetic progression. Recall that a sequence of real numbers is a real-valued function whose domain is the set of all natural numbers . An arithmetic sequence is a special type of sequence. It can be viewed as a function u: defined by u(n) = a + (n-1)b for some constants a, b ∈ . To shorten notation, we continue to write u_n instead of u(n). Here and subsequently, if an arithmetic sequence is defined in the form of u_n = a + (n-1)b as above, then a and b are called the first term and the common difference, respectively. For every , we call u_n the nth term of the sequence. The u_n‘s are called the values of the sequence. The value n itself is often called number of terms. It can be easily proved that u_n+1 – u_n = b for every .

Going back to the examples above, consider Sequence 1. The first term is 2, the second term is 5, the third term is 8, and so on. With the notation above, in Sequence 1, u₁ = 2, u₂ = 5, u₃ = 8, and so forth. Note that Sequence 1 has the following pattern: u₂ = u₁ + 3, u₃ = u₂ + 3, u₄ = u₃ + 3, and so on. In this case, u_n+1 = u_n + 3. It is a consequence of the essential property of an arithmetic sequence, u_n+1 – u_n = b (a constant) as described above. We can say that the sequence is identified by a = 2 and b = 3.

Sequence 2 is also an arithmetic sequence with the first term 11 and common difference 5. Thus, in this case a = 11 and b = 5. Likewise, Sequence 3 is characterized by a = 15 and b = -2.

Example 1

Find the 45^th term in the arithmetic sequence 4, 7, 10, 13, 16, …  Which term has the value 304?

Answer

In this example, a = 4 and b = 3. To determine the value of the 45^th term, substitute n = 45, a = 4, and b = 3 into the u_n formula above. This gives:

u₄₅ = 4 + (45 – 1)⋅3 = 4 + 44⋅3 = 4 + 132 = 136.

Thus, the value of the 45^th term is 136.

To determine which term has the value 304, let u_n = 304.

Note that u_n = 4 + (n-1)⋅3 = 3n + 1

As a consequence, 3n + 1 = 304, which is equivalent to n = 101.

In conclusion, the 101^th term has the value 304.

Example 2

How many natural numbers between 10 and 1000 are multiples of 3?

Answer

The smallest natural number between 10 and 1000 that is a multiple of 3 is 12. The largest natural number between 10 and 1000 that is a multiple of 3 is 999.

Consider the arithmetic sequence 12, 15, 18, 21, …, 999. It is characterized by a = 12 and b = 3. We have to find n satisfying u_n = 999. Note that u_n = a + (n – 1)⋅b = 12 + (n – 1)⋅3 = 3n + 9. Consequently, 3n + 9 = 999. This results in n = 330. Therefore, there are 330 natural numbers with the specified properties above.

Example 3

Eleven numbers must be inserted between 4 and 100 so that the first thirteen terms of a monotonic increasing arithmetic sequence are formed. Find the common difference of the corresponding sequence.

Answer

Let 4 be the first term of the sequence. Then, 100 will be the 13^th term, i.e. u₁₃ = 100. Note that a = 4 implies u₁₃ = 4 + (13 – 1)⋅b = 4 + 12b. Consequently, 4 + 12b = 100. This yields . So, the corresponding arithmetic sequence has a common difference of 8.

(to be continued)