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An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For instance, in the sequence 2, 4, 6, 8, ... the common difference is 2, as each term increments by this exact value. Identifying an arithmetic sequence is vital in various mathematical applications, especially when working with series and finding sums.

To determine if a set of numbers form an arithmetic sequence, one simply subtracts one term from the following term. If the result is the same for all consecutive pairs of terms, then the sequence is indeed arithmetic. This consistent step allows for the automation of calculations like sums, averages, and even predicting future terms with precision. Arithmetic sequences are everywhere in our daily lives – from the ticking of a clock to the arrangement of seats in a theater.

Summation of an arithmetic sequence can be efficiently determined using a particular formula: \(S = \frac{n}{2} * (F + L)\), where \(S\) represents the sum, \({n}\) the number of terms, \(F\) the first term, and \(L\) the last term. This formula is derived from the fact that the average of the first and last term (\(\frac{F + L}{2}\)) times the number of terms (\(n\)) will give the total sum.

For example, if you're asked to find the sum of the first ten natural numbers, 1 through 10, the formula makes it straightforward: You'd calculate \(S = \frac{10}{2} * (1 + 10) = 5 * 11 = 55\). This method eliminates the need for adding each term individually, thus saving time and reducing potential errors in calculation.

An odd integer sequence is a specific type of arithmetic sequence where each term is an odd number. The common difference between consecutive terms is always 2, since adding 2 to one odd number results in the next odd number. For example, 1, 3, 5, 7, ... is an odd integer sequence. When working with odd (or even) integer sequences, finding the sum involves being careful with the formula to ensure the sequence bounds are respected.

In the provided exercise, the sum of odd integers from 35 to 105 inclusive is sought. With each of these integers being odd, it's confirmed the sequence is indeed made of odd integers, increasing by 2 every step. Furthermore, such sequences also play a role in various real-world contexts, from coding algorithms that iterate over data to creating patterns in visual designs.