Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{n^{2}}{2 n+1}\\\ &a_{5}= \end{aligned}$$

Step-by-Step Solution

Answer

\(\frac{25}{11}\)

Step 1: Identify the general formula of the Sequence

The general formula for the given sequence is \(a_n = \frac{n^2}{2n+1}\). This formula gives the \(n^{th}\) term of the sequence.

Step 2: Substitute the Value

To find the \(5^{th}\) term, substitute \(n = 5\) in the general formula. So, \(a_5 = \frac{5^2}{2*5+1}\).

Step 3: Perform Simplification

Solve the right side of the equation. First calculate the numerator, \(5^2 = 25\), then calculate the denominator, \(2*5+1 = 11\).

Step 4: Calculate the Fifth Term

Now divide the numerator by the denominator. So, \(a_5 = \frac{25}{11}\).

Key Concepts

General Formula of a Sequence

Understanding sequences in algebra can feel like unraveling a mystery story. At the heart of every sequence is its general formula. For our sequence, the general formula is given by \(a_n = \frac{n^2}{2n+1}\). This formula allows us to find any term in the sequence, referred to as the \(n^{th}\) term.
In essence, the general formula is a blueprint. It tells us how to construct each term based on its position in the sequence (denoted by \(n\)). With this formula, you aren't just finding numbers, you're discovering how the sequence grows and evolves.
To use the general formula, simply replace \(n\) with the position of the term you want to find. This ability to replace \(n\) is a powerful tool that enables us to move from a broad description of the sequence to specific numerical terms.

Substitution in Algebra

Substitution is like putting a puzzle piece into its perfect spot. In mathematics, especially in algebra, substitution means replacing a variable with a specific value.
Let's look at our sequence example. We want to find the \(5^{th}\) term, so we substitute \(n = 5\) into our general formula \(a_n = \frac{n^2}{2n+1}\). This gives us \(a_5 = \frac{5^2}{2\cdot5+1}\).

• First, identify the term you need.
• Replace \(n\) with your chosen term number (here it is 5).
• Solve the equation step by step to avoid mistakes.

Substitution is a foundational skill in algebra that turns complex formulas into workable, solvable equations. It's about precision and care in replacing the right things, in the right places.

Simplification in Fractions

After substitution, the next journey is through simplification. Simplifying fractions can often feel like clearing up a cluttered room. Your goal is to make everything as neat and tidy as possible.
For our sequence, after substituting \(n = 5\), we have the fraction \(\frac{25}{11}\). This involves two main steps: simplifying the numerator and the denominator.

• Numerator: Calculate \(5^2\), resulting in 25.
• Denominator: Calculate \(2\cdot5+1\), resulting in 11.

With these calculations resolved, the fraction becomes \(\frac{25}{11}\), which is already in its simplest form. Simplification ensures we present results in their cleanest and most understandable format, emphasizing clarity and precision in mathematical communication.