Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{j=3}^{5} \frac{1}{j+1}$$

Step-by-Step Solution

Answer

Adding these values together gives a sum of \(\frac{1}{4} + \frac{1}{5} + \frac{1}{6} = \frac{15}{60} = \frac{1}{4}\)

Step 1: Substitute \(j\) = 3

Substitute \(j\) = 3 into the equation \(\frac{1}{j+1}\) giving \(\frac{1}{3+1} = \frac{1}{4}\)

Step 2: Substitute \(j\) = 4

Substitute \(j\) = 4 into the equation \(\frac{1}{j+1}\) giving \(\frac{1}{4+1} = \frac{1}{5}\)

Step 3: Substitute \(j\) = 5

Substitute \(j\) = 5 into the equation \(\frac{1}{j+1}\) giving \(\frac{1}{5+1} = \frac{1}{6}\)

Step 4: Sum the Values

Now sum all the values obtained: \(\frac{1}{4} + \frac{1}{5} + \frac{1}{6}\)

Key Concepts

Sequence and Series

In algebra, a sequence is a list of numbers ordered in a specific way, following a certain rule. It is like a roadmap that tells us where each number goes. Each number in the list is a term of the sequence. For example, in the exercise, the sequence defined by \( \frac{1}{j+1} \) starts with \( j = 3 \). Hence, the terms we get are \( \frac{1}{4}, \frac{1}{5}, \) and \( \frac{1}{6} \).

A series, on the other hand, is what you get when you add together the terms of a sequence. It's like converting our sequence into a single number, and this number gives us the sum of all the values. In the question, summing the sequence turns it into a series. The challenge is to add \( \frac{1}{4} \), \( \frac{1}{5} \), and \( \frac{1}{6} \) together to find the final sum.

Finding Sums

Finding the sum in a series can be approached step by step, especially when working with fractions. Imagine you have sliced a pizza into different-sized slices and you want to put them back together. When finding the sum of fractions, each fraction needs to have the same denominator so you can add them more easily.

• The exercise starts with substituting \( j \) into the equation \( \frac{1}{j+1} \) for given values to find each term. For \( j \) from 3 to 5, we get the terms \( \frac{1}{4}, \frac{1}{5}, \) and \( \frac{1}{6} \).
• The next step is to add these fractions. Since they have different denominators, we find a common denominator — the smallest number that all denominators can divide into without leaving a remainder.
• One common approach is to use the least common multiple (LCM) of the denominators 4, 5, and 6.
• The LCM of 4, 5, and 6 is 60. Convert each fraction to have a denominator of 60 and then add them.

Rational Numbers

Rational numbers are numbers that can be written as a fraction with an integer numerator and a non-zero integer denominator. They include both positive and negative fractions, as well as whole numbers, since these can be expressed as a fraction with a denominator of 1.

• In our exercise, all terms \( \frac{1}{4}, \frac{1}{5}, \frac{1}{6} \) are rational numbers.
• Rational numbers are important in calculations involving sequences and series because they provide exact values, unlike irrational numbers or approximations.
• Working with rational numbers also sharpens our skills in operations with fractions like finding common denominators and performing addition.

Understanding how to manipulate these numbers helps tackle not only arithmetic problems but also challenges in algebra and beyond.