Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$4,7,12,19,28, \dots$$

Step-by-Step Solution

Answer

The \(n\)th term of the sequence is given as: \((n + 1)^{2}-n\)

Step 1: Identify the pattern

First, compute the differences between consecutive terms. \(7 - 4 = 3\), \(12 - 7 = 5\), \(19 - 12 = 7\), \(28 - 19 = 9\). It's visible that the differences between the terms are consecutive odd numbers. This indicates the sequence might be related to squares, since the difference between consecutive squares is an odd number.

Step 2: Check the square relationship

Notice how each term in the sequence is 3 less than a perfect square: \(4 = 2^2 - 1, 7 = 3^2 - 2, 12 = 4^2 - 3, 19 = 5^2 - 4, 28 = 6^2 - 5\). This relationship continues, showing that each number in the sequence is one less than the square of the next integer.

Step 3: Formulate the nth term

Expressing the pattern found in the second step as formulas, the \(n\)th term of the sequence can be described as: \((n + 1)^{2}-n\)

Key Concepts

Difference of Squares

The "difference of squares" is a fascinating concept in mathematics. It typically refers to the formula \(a^2 - b^2 = (a + b)(a - b)\). However, in the context of sequences, it can appear in various forms that help us understand patterns in numbers.
In the given sequence, each term is derived as one less than a perfect square, like \(2^2 - 1\) for 4 or \(3^2 - 2\) for 7.
The difference between consecutive terms is made up of odd numbers. These differences are not just random odd numbers; they follow a repetitive increment, suggesting the sequence relates to perfect squares in a less direct way.
Such a pattern leads us to explore these differences, helping us form a strategy to find the nth term. Recognizing these relationships can turn a chaotic sequence into something that makes perfect sense.

Nth Term Formula

Finding the formula for the nth term is key to unraveling sequence patterns. It lets you calculate any term in the sequence without listing all previous terms.
For our sequence, after noting the square pattern, we identified that each term is the square of the number one higher than its position, minus that position number:
\((n + 1)^2 - n\).This formula effectively generalizes the sequence pattern we've discovered:

• For the first term (\(n=1\)), it gives \(2^2 - 1\) which is 3.
• For the second term (\(n=2\)), it gives \(3^2 - 2\) which is 7.

Using this formula, you can find any term directly by substituting the number of the term you're interested in (\(n\)) into the formula. This can be immensely helpful for analyzing or predicting patterns in large sets of data.

Consecutive Integers

"Consecutive integers" are numbers that follow each other in order, with each number differing by 1. They form the backbone of many sequences and mathematical formulas.
In our sequence, the key observation was that each term coefficients to consecutive numbers subtracted from squares:

• 4 is \(2^2 - 1\)
• 7 is \(3^2 - 2\)

Here, the consecutive integers (like 1, 2, ...,) are directly involved in forming terms, making understanding them pivotal for patterns.If you spot consecutive integers in your analysis, you can often identify a direct numerical relationship or pattern.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers where each term is generated or derived based on a rule or function. Recognizing patterns in these sequences is a fundamental skill in mathematics.
The sequence 4, 7, 12, 19, 28, ... demonstrates a blend of arithmetic and geometric progression elements:
The difference between consecutive terms is odd numbers in increasing order. But what truly defines it is a relation to squares.

• Each number is derived by fitting integers to a specific square minus a consistent number.
• This rule gives rise to an interesting mix of straightforward calculations and deeper insights into numerical patterns.

Recognizing and using these rules effectively demands practice and a keen eye for patterns. As you work through various sequences, you'll become more skilled in spotting and articulating these mathematical sequences.