Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum of the finite arithmetic sequence. Sum of the integers from -10 to 50

Step-by-Step Solution

Answer

The sum of this arithmetic sequence is 1220.

Step 1: Identify the first and last terms

The first term \(a_1\) is -10 and the last term \(a_n\) is 50.

Step 2: Calculate the number of terms

An integer sequence from -10 to 50 has 61 terms. This is because it includes both -10 and 50, therefore, the number of terms 'n' is 61.

Step 3: Use the formula to find the sum

Using the formula \(S = \frac{n}{2}(a_1 + a_n)\), plug in the values as such: \(S = \frac{61}{2}(-10 + 50)\).

Key Concepts

Finite Sequence

A finite sequence is a collection of numbers listed in a specific order that has both a defined first element and a last element. In other words, it is a sequence with a limited number of items. The numbers in a sequence are often referred to as terms. In our example, the sequence starts with -10 and ends with 50, establishing a clear beginning and end.
Finite sequences are handy in various calculations because you can determine exact values without needing infinite terms. It allows us to manipulate and work with the sequence to find sums and other useful information.

Sum of Integers

The sum of integers for a sequence is the result of adding all the terms in the sequence together. When dealing with a finite arithmetic sequence, there is a particular method to calculate this sum efficiently.
In our exercise, we were asked to find the sum of integers from -10 to 50. To achieve this, we did not simply add each number one by one; instead, we used a formula designed for arithmetic sequences. This approach condenses the work and reduces the chance of calculation errors.
By applying the sequence sum formula, you can compute the result quickly and accurately.

Number of Terms

The number of terms in a sequence represents how many numbers are included from the start to the finish of the sequence. Determining the number of terms is crucial since it allows us to correctly use formulas that incorporate this information.
In our problem, we calculated the number of terms between -10 and 50. To find this, count each integer starting from -10 up to and including 50.A quick way to calculate it is by the formula: \(n = (\text{last term} - \text{first term}) + 1\). This yields 61 terms in our case because it accounts for both the first and last terms themselves as part of the sequence.

Sequence Formula

The sequence formula for finding the sum of an arithmetic sequence is a powerful tool. It allows us to swiftly and accurately calculate the sum without laborious adding.
The formula used is:

• \( S = \frac{n}{2}(a_1 + a_n) \)

Where:

• \(S\) is the sum of the sequence.
• \(n\) is the number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the last term.

In the context of our sequence from -10 to 50, this formula gave us \(S = \frac{61}{2}(-10 + 50)\).
This formula is particularly significant for arithmetic sequences as it uses the properties of uniform spacing between terms to simplify your calculations drastically.