Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$1,3,1,3,1, . . .$$

Step-by-Step Solution

Answer

The nth term of the sequence can be represented as \(a_n = 2 - (n \mod 2)\).

Step 1: Identify the pattern

Observe the given sequence: 1,3,1,3,1,... It's clear that the sequence is alternating between 1 and 3. In other words, if 'n' is an odd number the nth term is 1 and if 'n' is an even number the nth term is 3.

Step 2: Express the pattern mathematically

Based on our observation, we can express the nth term of the sequence as such: a_n = 2 if n is even, and a_n = 1 if n is odd.

Step 3: Rewrite the pattern using modular arithmetic

To get a more compact expression for the nth term, we can use modular arithmetic. Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value - the modulus. Rewriting the pattern in this format gives \(a_n = 2 - (n \mod 2)\).

Key Concepts

Mathematical Patterns

Mathematical patterns are like the secret codes of mathematics. They're all around us, giving structure to seemingly random numbers. Think of them as a way to predict what number comes next in a sequence. In our alternating sequence example, the pattern is a simple dance between the numbers 1 and 3. This back and forth is so regular that you can predict the numbers that will come at any position.

Patterns are not just about prediction, though; they help in understanding the intrinsic properties of numeric sequences. Identifying patterns can lead to developing formulas or expressions that describe the entire sequence in a compact form. This not only allows us to comprehend how the sequence behaves but also enables us to communicate large amounts of information efficiently. It's like knowing the recipe rather than listing every ingredient each time you cook a particular dish.

Modular Arithmetic

Picture modular arithmetic as a clock. Just as a clock wraps around from 11 to 12, then back to 1, modular arithmetic wraps around after hitting a certain number, called the modulus. If we talk about mod 2, like in our sequence, we're essentially considering a world where numbers reset after 2, much like a toggle switch.

In the context of our alternating sequence, we use mod 2 to distinguish between even and odd terms. When we take any number 'n' and perform modulo 2 arithmetic (written as \(n \mod 2\)), we're left with either 0 if 'n' is even, or 1 if 'n' is odd. So, applying modular arithmetic to the nth term of the sequence ties back to the original pattern we observed and provides us with a clever and elegant way to express it.

Sequence Expressions

A sequence expression is like a wizard's spell – it's a formula that conjures up any term in the sequence without the need to look at the entire sequence itself. For the sequence at hand, we saw that the nth term alternates between 1 and 3. Instead of writing out the entire sequence or using two separate expressions for odd and even 'n', we use an expression that encapsulates the entire sequence succinctly.

The expression \(a_n = 2 - (n \mod 2)\) is one such incantation. It cleverly leverages the result of modular arithmetic to alternate between 1 and 3. When 'n' is even, \(n \mod 2\) equals 0, so \(a_n = 2 - 0 = 2\). When 'n' is odd, \(n \mod 2\) equals 1, and therefore \(a_n = 2 - 1 = 1\). Voilà, we have our magic formula that reveals the value of any term at the wave of a mathematical wand!