Problem 35 - Precalculus with Limits

Question

In Exercises 33-36, find the indicated term of the sequence. \( a_n = \dfrac{4n}{2n^2 - 3} \) \( a_{11} = \Box \)

Step-by-Step Solution

Answer

The 11th term of the sequence \( a_{11} \) is \( \dfrac{44}{239} \).

Step 1: Substitute the given value

We will start by substituting the n in the given rule with 11, which is the indicated term's position: \( a_{11} = \dfrac{4*11}{2*11^2 - 3} \).

Step 2: Simplify the expression

Next, we would calculate the expressions at the numerator and the denominator of the fraction separately, then divide them. This simplifies to \( a_{11} = \dfrac{44}{2*121 - 3} = \dfrac{44}{242-3} = \dfrac{44}{239} \).

Step 3: Final Evaluation

Lastly, we would carry out the division of the two numbers in the fraction to get the value of \( a_{11} \). Or, if the division can't be performed exactly, the fraction is kept as it is.

Key Concepts

Term of a Sequence

In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in this list is called a 'term.' The position of a term in a sequence is denoted by a subscript "n."
For example, the term at position 11 in a sequence is denoted as \( a_{11} \). The sequence's rule, or formula, allows you to calculate any term as long as you know its position. This is very important because once you know the formula, you can find any term's value without having to list all previous terms.
In this exercise, the rule for the sequence is given by the formula \( a_n = \dfrac{4n}{2n^2 - 3} \). We use this to find the value of the 11th term \( a_{11} \) by plugging 11 into the formula.

Substitution Method

The substitution method is a powerful tool in algebra, especially useful in finding the value of a sequence's term. You substitute the position number into the formula given for the sequence.
In this exercise, to find \( a_{11} \), we substitute \( n = 11 \) in the formula \( a_n = \dfrac{4n}{2n^2 - 3} \).
This gives us:

• Numerator: \( 4 \times 11 = 44 \)
• Denominator: \( 2 \times 11^2 - 3 \)

This method allows us to find the exact mathematical expression that represents \( a_{11} \) in our sequence.

Numerator and Denominator Simplification

Simplifying the numerator and the denominator is crucial when working with fractions. It reduces complexity and makes equations easier to manage.
For the given sequence, \( a_n = \dfrac{4n}{2n^2 - 3} \), we calculated:

• Numerator as \( 44 \)
• Denominator as \( 2 \times 121 - 3 = 239 \)

Simplification here means ensuring the equation represents the simplest form possible before dividing. For any given fraction, it's easier to work with smaller numbers. Making calculations less overwhelming and reducing room for errors.

Fraction Division

Division in fractions means finding how many times the denominator fits into the numerator. However, sometimes the result isn't a perfect number. In such cases, it's common to leave the result as a simplified fraction.
For \( a_{11} = \dfrac{44}{239} \), it's important to check if this can further be simplified or expressed as a decimal. If exact division isn't possible or doesn't result in a simple fraction, it's perfectly acceptable to leave it in this fractional form.
This method highlights precision and the elegance of mathematics, where not every problem leads to a whole number answer, but understanding the concept remains crucial.