Problem 37 - College Algebra

Question

Find the sum of the first 50 terms of the arithmetic sequence: \(-10,-6,-2,2, \dots\)

Step-by-Step Solution

Answer

The sum of the first 50 terms of the given arithmetic sequence is 3400.

Step 1: Find the First Term (a1)

The first term of the sequence is \(a_1 = -10\)

Step 2: Calculate the Common Difference (d)

The common difference could be found as the difference between any two successive terms. Here, the second term is -6 and the first term is -10, so difference \(d = -6 - (-10) = 4\)

Step 3: Find the 50th Term (a50)

The 50th term of an arithmetic sequence can be found using the formula \(a_n = a_1 + (n-1) * d\). Substituting the known values in, \(a_{50} = -10 + (50-1) * 4 = 146\)

Step 4: Calculate Sum of the terms

The sum of the first 50 terms can be calculated using formula \(S_n = \frac{n}{2} * (a_1 + a_n)\). Substituting the known values in, \(S_{50} = \frac{50}{2} * (-10 + 146) = 50 * 68 = 3400\)

Key Concepts

Arithmetic Sequence Sum

Understanding the summation of an arithmetic sequence is key when dealing with linear patterns in numbers. An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant, known as the common difference. In the exercise, we find a sequence starting with -10 and increasing by 4 each time.

To find the sum of the first 'n' terms of such a sequence, one effective formula is \(S_n = \frac{n}{2} * (a_1 + a_n)\). This formula is derived from the concept that the sum of an arithmetic series is the average of the first and last term, multiplied by the number of terms. In the case of the first 50 terms of the given sequence, this results in a sum of 3,400. It's like finding the average of the first and last number and then realizing the rest of the numbers fit symmetrically in between to give us our total sum.

Common Difference

A key characteristic of an arithmetic sequence is its common difference 'd', which is the consistent interval between any two consecutive terms. To find the common difference, simply subtract one term from the next. For instance, with the terms -10 and -6, the common difference is \(d = -6 - (-10) = 4\).

This difference remains the same throughout the sequence, and it is this steady step that allows an arithmetic sequence to be predictable and hence easy to work with. The common difference not only helps in identifying an arithmetic sequence but is also instrumental when finding other terms in the sequence or calculating the sum of a certain number of terms.

Sequence Term Calculation

Calculating individual terms in an arithmetic sequence is straightforward once the first term and the common difference are known. The general formula to find the nth term, \(a_n\), in an arithmetic sequence is \(a_n = a_1 + (n-1) * d\). This formula embodies the pattern by stating that to reach the nth term, one should start at the first term \(a_1\) and add the common difference 'd' a specific number of times, which is one less than 'n'.

In our case, to find the 50th term, we calculate \(a_{50} = -10 + (50-1) * 4\), which gives us 146. This demonstrates how an arithmetic sequence grows and how any term within the sequence can be pinpointed without listing out all terms before it.