Problem 38 - Algebra for College Students

Question

Find the sum of the first 60 terms of the arithmetic sequence \(-2,3,8,13,18, \ldots 8730\)

Step-by-Step Solution

Answer

The sum of the first 60 terms is 8730.

Step 1: Identify the First Term

The first term of the arithmetic sequence is denoted by \( a_1 \). From the given sequence, we can see that \( a_1 = -2 \).

Step 2: Identify the Common Difference

The common difference \( d \) is found by subtracting the first term from the second term. In this sequence, \( d = 3 - (-2) = 5 \).

Step 3: Determine the General Formula

The general term \( a_n \) of an arithmetic sequence can be found using the formula: \( a_n = a_1 + (n-1)d \). Substituting the known values gives us \( a_n = -2 + (n-1) \times 5 \).

Step 4: Calculate the 60th Term

Substitute \( n = 60 \) in the formula for \( a_n \) to find the 60th term. \( a_{60} = -2 + (60-1) \times 5 = -2 + 295 = 293 \).

Step 5: Use the Sum Formula for Arithmetic Series

The sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \). Substituting the values: \( n = 60, a_1 = -2, a_{60} = 293 \), we get: \( S_{60} = \frac{60}{2}(-2 + 293) = 30 \times 291 = 8730 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the constant amount that each term increases from the one before it. It determines the rate of change between consecutive terms. To find the common difference (\( d \)), simply subtract the first term of the sequence from the second term. For example, in the sequence \(-2, 3, 8, 13, ...\), we see that each term increases by 5 from the previous term. So, the common difference \( d = 3 - (-2) = 5 \). This shows that every step along the sequence adds a fixed number, making calculations predictable and consistent.The constant nature of the common difference is crucial because it allows us to apply formulas and find specific terms or even the sum goals in an efficient and straightforward manner. If the common difference changes, the sequence wouldn't be arithmetic anymore.

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. When the terms of an arithmetic sequence are added together, it forms a series. Understanding this concept is essential as it extends the idea of individual terms in a sequence to their collective sum. Consider an arithmetic sequence like \(-2, 3, 8, 13, ...\) . If we take a defined number of these terms, say the first 60 terms, and add them all together, we achieve the arithmetic series.It's important to grasp that while sequences deal with individual terms, a series represents the summation of those terms. The formula for the sum of an arithmetic series helps in easily calculating the sum without adding each term manually.

Sum of Sequence

The sum of an arithmetic sequence can be efficiently calculated using the formula for the sum of an arithmetic series. This sums up a specified number of terms, making it useful for problems involving long sequences where manual addition isn't practical. The formula to find the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]where \( a_1 \) is the first term, \( a_n \) is the last term, and \( n \) is the number of terms.Using the sequence from our example, to find the sum of the first 60 terms where \( a_1 = -2 \) and the 60th term \( a_{60} = 293 \), we plug these values into the formula:\[ S_{60} = \frac{60}{2}(-2 + 293) = 30 \times 291 = 8730.\]The formula is a quick way to find the sum and is often used in mathematics for various applications involving series.