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Chapter 10: Problem 27

Find the sum. Sum of the odd integers from 5 to \(125,\) inclusive

Short Answer

Expert verified
The sum of all the odd integers from 5 to 125 is 3960.

Step by step solution

01

Identify the sequence

Firstly, identify the odd integers from 5 to 125 inclusive. Odd integers are numbers that have a remainder of 1 when divided by 2. The first term (a) is 5 and the last term (l) is 125.
02

Find the common difference

In an arithmetic sequence of odd or even numbers, the common difference (d) is always 2. That is because the difference between consecutive odd or even numbers is always 2.
03

Determine the number of terms

The number of terms (n) in the sequence can be found using the formula \(n = \frac{{l - a}}{{d}} + 1\). In this case, \(n = \frac{{125 - 5}}{{2}} + 1 = 61\)
04

Apply the arithmetic series formula

Sum of the terms of an arithmetic progression is given by \(S_n = \frac{n}{2} \cdot (a + l)\). Substitute the values \(n = 61\), \(a = 5\), \(l = 125\) into the formula to get the sum \(S_n = \frac{61}{2} \cdot (5 + 125) = 3960\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Odd Integers
Odd integers are a type of whole number. They are characterized by having a remainder of 1 when divided by 2. For instance, numbers like 1, 3, 5, 7, and so on, are odd integers. This is because when you divide them by 2, the leftover part is 1. Odd integers appear regularly, alternating with even numbers, which are exactly divisible by 2. When identifying a sequence of odd integers, like from 5 to 125 as seen in our exercise, it's important to ensure that every number in the sequence follows this rule. Here, both 5 and 125, the first and last terms, are odd integers, confirming the sequence is correctly defined.
Common Difference in Arithmetic Sequences
The concept of a common difference is central to arithmetic sequences. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. This difference is known as the common difference. For sequences of odd numbers, this difference is consistent and always equals 2. Why is this the case? Because subtracting any odd number from the next closest one always gives 2. For instance, in the sequence given in the exercise 鈥� 5, 7, 9, 11, ..., 125 鈥� each successive odd number is 2 units more than the previous one. This consistency helps easily identify arithmetic sequences and calculate related tasks.
Arithmetic Series Formula
Calculating the sum of terms in an arithmetic sequence is simplified by using the arithmetic series formula. The formula you need is \( S_n = \frac{n}{2} \cdot (a + l) \). Here, \( S_n \) represents the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. In our example, where the odd numbers are from 5 to 125, we have determined that there are 61 terms. Plugging in these values鈥擻( a = 5 \), \( l = 125 \), and \( n = 61 \) 鈥� into the formula we find: \( S_n = \frac{61}{2} \cdot (5 + 125) = 3960 \). This method makes it straightforward to handle and compute sums efficiently in similar problems.

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