Problem 51 - Algebra and Trigonometry

Question

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ 0.7+2.7+4.7+\cdots+56.7 $$

Step-by-Step Solution

Answer

The sum is 832.3.

Step 1: Identify parameters of the arithmetic sequence

In the given sequence, the first term \(a_1\) is 0.7, and the common difference \(d\) is the difference between 2.7 and 0.7, which is 2.0. We need to find the sum of this arithmetic series where the last term \(a_n\) is 56.7.

Step 2: Find the number of terms \(n\)

The \(n\)-th term of an arithmetic sequence is given by the formula \(a_n = a_1 + (n-1) \times d\). Substitute \(a_n = 56.7\), \(a_1 = 0.7\), and \(d = 2.0\) into the formula: \[56.7 = 0.7 + (n-1) \times 2.0\] Simplifying gives: \[56.7 - 0.7 = 2.0(n-1)\] \[56 = 2.0(n-1)\] \[28 = n-1\] \[n = 29\] There are 29 terms in the sequence.

Step 3: Use the sum formula for an arithmetic series

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2} \times (a_1 + a_n)\). Substitute \(n = 29\), \(a_1 = 0.7\), and \(a_n = 56.7\) into the formula: \[S_{29} = \frac{29}{2} \times (0.7 + 56.7)\] \[S_{29} = \frac{29}{2} \times 57.4\] \[S_{29} = 29 \times 28.7\] \[S_{29} = 832.3\] The sum of the sequence is 832.3.

Key Concepts

Understanding Partial Sum

When you hear the term "partial sum" in relation to an arithmetic sequence, it refers to the sum of a first few terms in that sequence. In our case, it means we're adding up all the numbers from the start of the sequence up to a specified term.
For example, in the sequence given: 0.7, 2.7, 4.7, ..., 56.7, we want to find the sum from the first term up to the term that equals 56.7. This is not to be confused with summing the *entire* sequence if it doesn't end at this term.
By the end of the exercise, understanding partial sums helps us conclude that the sum of these specific terms provides a focused result of 832.3. It's important in practical applications, too, like financial forecasting or data analysis, where one might need to know the sum of certain data points rather than all available data. 🚀

Exploring Common Difference

In any arithmetic sequence, a consistent difference between consecutive terms appears, known as the "common difference." This value is crucial because it dictates the pace at which the sequence progresses.
In our example of 0.7, 2.7, 4.7, ..., 56.7, the common difference is 2.0. We can find this by subtracting the first term from the second, or:

• Common difference = 2.7 - 0.7 = 2.0

This difference creates a predictable pattern.

• The sequence jumps 2 units from one term to the next.

Whether you're analyzing data or working on mathematical problems, knowing the common difference helps you pinpoint future terms and verify existing ones. Understanding how each term increases or decreases essentially frames your entire sequence in a clear light 📈.

The Role of Arithmetic Series

An arithmetic series is essentially the expression of adding up all the terms of an arithmetic sequence. Think of a series as a way to summarize a long list of numbers into a single total.
In the context of our example, the sequence of terms leads to the arithmetic series sum formula:

• Sum of the series = \( S_n = \frac{n}{2} \times (a_1 + a_n) \)
• For the series 0.7 to 56.7, the total we calculated was \( 832.3 \).

Using this series concept, the contribution of each term becomes significant, showing that the structured summation can have practical applications, like calculating total amounts in statistical data or accumulating expenses over a set period of time. Understanding this concept bridges sequences to more extensive applications. 💡

Determining the Number of Terms

Finding the number of terms in an arithmetic sequence is pivotal, as it influences how we calculate its total sum. In this sequence,

• Starting at 0.7,
• Ending at 56.7,
• With a common difference of 2.0,

we must identify how many terms connect these bounds. The formula for the nth term of an arithmetic sequence helps us unravel this equation: \[ a_n = a_1 + (n-1) imes d \] In our case, solving, we found:

• \( n = 29 \), meaning there are 29 terms.

With this understanding, we can efficiently sum the sequence. Knowing the number of terms reflects not only our math skill but aids in practical situations like determining the number of payments for loans or calculating mile markers on travel routes. 🛤️