Problem 39 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(0,1,4,9,16, \dots\)

Step-by-Step Solution

Answer

The general term formula is a_n = (n-1)^2.

Step 1 - Analyze the Sequence

Observe the given sequence: 0, 1, 4, 9, 16,... Notice that each term in the sequence can be related to the square of its position in the sequence.

Step 2 - Identify the Pattern

Identify the relationship between the term's position, denoted as n (starting from n=1 for the first term), and the value of the term. Here, each term seems to be the square of n-1.

Step 3 - Write the General Term

Write the general term formula based on your observations. If n is the position in the sequence starting from 1, the corresponding term is Therefore, the nth term is: a_n = (n-1)^2

Key Concepts

infinite sequence

An infinite sequence is a sequence that continues endlessly. In mathematics, such sequences do not stop at a certain number but extend indefinitely.
Infinite sequences are often used to describe patterns or sets of numbers that go on forever.
For example, the sequence 0, 1, 4, 9, 16,... is an infinite sequence of square numbers. Each number in the sequence has a specific position, or index, and the sequence can be described using a general formula.

problem-solving steps

Approaching a problem systematically is key to finding the correct solution. Here are some steps to solve the problem of finding the general term of an infinite sequence:

• Analyze the Sequence: Look at the given terms to find a pattern or relationship. For example, in the sequence 0, 1, 4, 9, 16,..., notice how each term grows as the sequence progresses.
• Identify the Pattern: Determine how each term in the sequence relates to its position. In our example, each term is the square of its position minus one.
• Write the General Term: Based on the pattern, write a formula that represents any term in the sequence. For the given sequence, the formula for the nth term is \(a_n = (n-1)^2\).

Using these steps can help break down complex problems into more manageable parts.

square numbers

Square numbers are the result of multiplying a number by itself. The sequence 0, 1, 4, 9, 16,... is a classic example of square numbers.
To better understand square numbers:

• 0 is the square of 0: \(0^2 = 0\)
• 1 is the square of 1: \(1^2 = 1\)
• 4 is the square of 2: \(2^2 = 4\)
• 9 is the square of 3: \(3^2 = 9\)
• 16 is the square of 4: \(4^2 = 16\)

Square numbers grow larger much faster than the natural numbers, and each term in the sequence represents a square of its predecessor's position plus one. Recognizing this pattern can help in forming the general term for the sequence, enhancing problem-solving abilities in mathematics.