Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

$$1+\frac{1}{1}, 1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \dots$$$$1+\frac{1}{1}, 1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \dots$$

Step-by-Step Solution

Answer

The nth term in the sequence is given by the formula \(T_n = 1 + \frac{1}{n}\).

Step 1: Identify the pattern

The given sequence is \(1 + \frac{1}{1}\), \(1+ \frac{1}{2}\), \(1+ \frac{1}{3}\), \(1+ \frac{1}{4}\), \(1 + \frac{1}{5}\), ... . We can see in each term, the numerator is always 1, and the denominator corresponds to the term number.

Step 2: Evaluate each term

To find the value of each term, we simply add 1 to the reciprocal of the term number. So, for the first term, we add 1 to the reciprocal of 1, which equals 2. For the second term, we add 1 to the reciprocal of 2, which equals 1.5. And so on. We continue this process for all terms.

Step 3: Generalize the term

We can simplify this to a formula that gives us the value of any term in the sequence. The nth term in this sequence, \(T_n\), is given by \(T_n = 1 + \frac{1}{n}\). So to find the 5th term, for example, we plug 5 in for n in the formula to get \(T_5 = 1 + \frac{1}{5} \) or 1.2.

Key Concepts

Reciprocal

Understanding the concept of a reciprocal is essential when dealing with sequences, especially ones involving fractions. The reciprocal of a number is simply one divided by that number. Whenever we have a number \( n \), its reciprocal is expressed as \( \frac{1}{n} \). This means if the number is 2, its reciprocal is \( \frac{1}{2} \), which results in 0.5 when converted into a decimal.

Reciprocals play a significant role in the sequence given in our exercise. Each term involves adding 1 to the reciprocal of the term's position in the sequence. For example, the first term in the sequence is obtained by adding 1 to \( \frac{1}{1} \), which computes to 2. Understanding reciprocals helps us break down seemingly complex fractions into simple numerical values.

Nth term

The formula for the nth term of a sequence is a powerful tool that allows us to find any term's value without having to calculate each preceding term.

In the given sequence, an easy pattern emerges where each term is calculated by adding 1 to the reciprocal of its position number. We derive the formula \( T_n = 1 + \frac{1}{n} \). Let's break it down:

• "1" remains constant in each term, serving as the base value.
• "\( \frac{1}{n} \)" changes with each term and reflects the reciprocal of the term's position.

This formula represents a straightforward way to calculate any specific term in the sequence. If you wanted to find the 10th term, replace \( n \) with 10, so \( T_{10} = 1 + \frac{1}{10} \), which equals 1.1. This shows how the formula adapts quickly and efficiently for any required term.

Pattern recognition

When studying sequences, recognizing patterns is an essential skill that helps make sense of the numbers and their progression. Observing the sequence provided, we notice a repetitive pattern in the way each term is constructed.

• The numerator of each added fraction is consistently 1 in every term.
• The denominator corresponds exactly with the term's position number."

Through this observation, we can establish that this sequence forms a recognizable pattern of \( 1 + \frac{1}{n} \), as demonstrated during the solution process. Recognizing patterns simplifies the calculation process, makes it easier to identify rules governing the sequence, and helps construct formulas like the nth term. Such skills are valuable in diagnosing and understanding math problems efficiently.