Problem 29 - Thomas Calculus

Question

Find a formula for the \(n\)th term of the sequence. $$ \sin \left(\frac{\sqrt{2}}{1+4}\right), \sin \left(\frac{\sqrt{3}}{1+9}\right), \sin \left(\frac{\sqrt{4}}{1+16}\right), \sin \left(\frac{\sqrt{5}}{1+25}\right), \dots $$

Step-by-Step Solution

Answer

\( a_n = \sin\left(\frac{\sqrt{n+1}}{n^2 + 2n + 2}\right) \)

Step 1: Identify the General Term Structure

Observe that each term of the sequence is expressed in the form \( \sin\left(\frac{\sqrt{k}}{1+k^2}\right) \). To identify the function of \( n \), notice that the value under the square root in each term increases by 1 with each subsequent term.

Step 2: Express k in terms of n

Let's find a relation between \( k \) and \( n \). Start by comparing the sequence terms to natural numbers: \( k = 2, 3, 4, 5, \ldots \). Notice that \( k = n + 1 \). Thus, \( k = n + 1 \) is a function that relates each sequence element to its natural number position \( n \).

Step 3: Substitute k in the General Formula

Substitute \( k = n+1 \) into the general expression \( \sin\left(\frac{\sqrt{k}}{1+k^2}\right) \). The nth term of the sequence is then \( \sin\left(\frac{\sqrt{n+1}}{1+(n+1)^2}\right) \).

Step 4: Simplify the Expression

Simplify \( 1+(n+1)^2 \) as follows: \( 1+(n+1)^2 = 1 + (n^2 + 2n + 1) = n^2 + 2n + 2 \). So, the nth term formula simplifies to \( \sin\left(\frac{\sqrt{n+1}}{n^2 + 2n + 2}\right) \).

Key Concepts

Trigonometric Sequence

A trigonometric sequence is a sequence that involves trigonometric functions like sine, cosine, and tangent. In our exercise, the sequence uses the sine function to create each term. This sequence starts with the expression \( \sin\left(\frac{\sqrt{2}}{1+4}\right) \).
This kind of sequence is interesting because it combines trigonometric functions with algebraic expressions. Each term in this sequence results from a sine function involving a fraction. The numerator of this fraction is the square root of an increasing number, while the denominator involves squaring these increasing numbers. Such sequences can have fascinating mathematical properties, often oscillating or repeating in a predictable pattern.

General Term

The general term in a sequence allows us to express any term based on its position, \( n \), in the sequence. Think of it as a formula that tells you exactly how to find the nth term. In this exercise, each term is structured as \( \sin\left(\frac{\sqrt{k}}{1+k^2}\right) \) where \( k \) is adjusted for each \( n \).
By identifying that \( k = n + 1 \), we can substitute this into the sequence term to specify that the nth term becomes \( \sin\left(\frac{\sqrt{n+1}}{n^2 + 2n + 2}\right) \). The concept of the general term is crucial as it simplifies the process of finding any specific term within the sequence without listing all preceding terms.

Sequence Formula

The sequence formula is the mathematical expression that gives us each term's value in the sequence based on its position number, \( n \). In our scenario, we derived the formula \( \sin\left(\frac{\sqrt{n+1}}{n^2 + 2n + 2}\right) \).
To find this, we first analyzed the structure of the terms in our sequence. The denominator \( 1 + k^2 \) was particularly informative, as it transformed into \( n^2 + 2n + 2 \) when substituting \( k = n + 1 \). This kind of formula is particularly useful as it provides a direct way to compute any term directly rather than expanding or exploring previous values.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific rule or pattern. Each element of a sequence is related by a consistent rule, expressed through a formula or function. They can be arithmetic, geometric, or, as in our example, involve more complex functions such as trigonometric ones.
Mathematical sequences are foundational in various fields of mathematics. They help in understanding convergence, growth patterns, and can be solutions to complex problems. In the case of the sinusoidal sequence we're considering, we see how combining arithmetic operations (like addition and squaring) with trigonometric functions forms a unique pattern, illustrating the broad scope of mathematical sequence types.