Problem 57 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. Sum of the integers from \( -100 \) to \( 30 \)

Step-by-Step Solution

Answer

The sum of the integers from -100 to 30 is -4555.

Step 1: Identify the Given Values

Here we have -100 as the first integer and 30 as the last integer of the sequence, making \( a = -100 \) and \( l = 30 \). Moreover, we have that the arithmetic sequence is the set of all integers from -100 to 30, so the number of terms \( n \) is given by \( n = l - a + 1 = 30 - (-100) + 1 = 131 \).

Step 2: Apply the formula for the sum of an arithmetic sequence

Substitute \( a = -100 \), \( l = 30 \), and \( n = 131 \) into the formula \( S = \frac{n}{2} (a + l) \). After applying these values, we get \( S = \frac{131}{2} (-100 + 30) \).

Step 3: Compute the sum of the sequence

Carry out the arithmetic in \( S = \frac{131}{2} (-100 + 30) \) to obtain the sum \( S = -4555 \).

Key Concepts

Sum of a Sequence

When it comes to understanding how to find the sum of a sequence, especially in arithmetic progressions, this notion is core. A sequence is simply a list of numbers following a particular order or pattern. In arithmetic sequences, each term after the first is obtained by adding a fixed number, known as the common difference, to the previous term.
To find the sum of such a sequence, you add together all of its terms. This operation can be quite intuitive when dealing with a small number of terms but when faced with larger sequences, systematic approaches become invaluable.
The sum of a sequence can be efficiently computed by using the appropriate formula for its type, which leads us to the arithmetic series formula, a powerful tool to simplify this task.

Arithmetic Series Formula

The arithmetic series formula is a key tool when summing terms in an arithmetic sequence. This formula accelerates the process by avoiding the need to add numerous terms manually.
The arithmetic series formula is given by \[ S = \frac{n}{2} (a + l) \] where:

• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

This formula essentially calculates the average of the first and last terms and multiplies it by the number of terms. This simplification is particularly handy because it only requires the basic knowledge of the first and last terms and the total number of terms in the sequence.
For example, when summing integers from -100 to 30, the formula helps swiftly determine that the sum is -4555, as shown by substituting our known values into the formula.

Finite Sequences

Finite sequences are sequences that have a limited number of terms. Unlike infinite sequences that stretch on without end, finite sequences conclude after a specific number of elements.
In arithmetic sequences, this concept is important because it directly relates to the way we sum them. Being finite means we can exactly count how many terms are in the sequence, a requirement for using the arithmetic series formula.
For example, in the problem of finding the sum of integers from -100 to 30, you need to establish that there are exactly 131 terms from the beginning to the end of the sequence. This understanding comes from calculating how many times the fixed common difference is added to reach from the first term to the last, providing clarity and precision in solving such problems.