Problem 41 - Contemporary Precalculus

Question

Find the kth partial sum of the arithmetic sequence \(\left\\{a_{n}\right\\}\) with common difference d. $$k=6, a_{1}=2, d=5$$

Step-by-Step Solution

Answer

Answer: The 6th partial sum of the given arithmetic sequence is 87.

Step 1: Find the 6th term of the arithmetic sequence

First, let's find the nth term of the arithmetic sequence using the formula: $$a_n = a_1 + (n - 1)d$$ Plug in the given values, \(n = 6\), \(a_1 = 2\), and \(d = 5\): $$a_6 = 2 + (6 - 1)(5) = 2 + (5)(5) = 2 + 25 = 27$$ So, the 6th term of the arithmetic sequence is 27.

Step 2: Calculate the 6th partial sum of the arithmetic sequence

Now that we have the 6th term, let's calculate the 6th partial sum using the formula: $$S_k = k(\frac{a_1 + a_n}{2})$$ Plug in the values, \(k = 6\), \(a_1 = 2\), and \(a_6 = 27\): $$S_6 = 6(\frac{2 + 27}{2}) = 6(\frac{29}{2}) = 3(29) = 87$$ So, the 6th partial sum of the arithmetic sequence is 87.

Key Concepts

nth term formula

To understand the nth term formula and its importance in arithmetic sequences, let's start with the basics. The nth term formula is a way to find any term in an arithmetic sequence, without needing to list all previous terms. This formula is given by \[a_n = a_1 + (n - 1) imes d\] where:

• \(a_n\) is the term you want to find,
• \(a_1\) is the first term of the sequence,
• \(n\) is the term number, and
• \(d\) is the common difference between any two consecutive terms.

In our original exercise, with \(a_1 = 2\), \(d = 5\), and \(n = 6\), the formula helps us find that the 6th term (\(a_6\)) is 27. This formula is powerful because it can be used to quickly identify any term in an arithmetic pattern, making it a critical tool in sequence analysis.

partial sum

The concept of partial sums in arithmetic sequences is important for calculating the sum of a specific number of initial terms in the sequence. A partial sum is simply the sum of the first \(k\) terms of an arithmetic sequence. The formula to calculate the partial sum, denoted as \(S_k\), is:\[S_k = k \left( \frac{a_1 + a_n}{2} \right)\]where:

• \(S_k\) is the partial sum of the first \(k\) terms,
• \(k\) is the number of terms,
• \(a_1\) is the first term, and
• \(a_n\) is the nth term.

In our scenario, calculating the 6th partial sum involves substituting \(a_1 = 2\), \(a_6 = 27\), and \(k = 6\) into the formula, giving us a sum of 87. Understanding partial sums is key for analyzing patterns or trends within sections of larger sequences.

common difference

In any arithmetic sequence, the common difference is what makes each term in the sequence increase or decrease by a constant amount. It is defined as the difference between consecutive terms, mathematically represented as:\[d = a_2 - a_1\]where:

• \(d\) is the common difference,
• \(a_2\) is the second term, and
• \(a_1\) is the first term.

For instance, in our exercise where the common difference \(d = 5\), the sequence increases by 5 every step. Thus, it plays a vital role in determining how the sequence develops over time. Finding the common difference is often one of the first steps in understanding the sequence, as it allows you to construct further terms easily.

sequence analysis

Sequence analysis involves examining the structure and properties of number sequences. When dealing with arithmetic sequences in particular, several insights can be drawn:

• The trend or pattern of change between terms, highlighted by the consistent common difference \(d\).
• The ability to predict future terms with the nth term formula.
• Calculation of partial sums to understand the cumulative growth or decline across terms.

The example exercise demonstrates how to find specific terms using given values and formulas, enabling one to construct parts of the sequence without difficulty. Sequence analysis is valuable in solving practical problems involving regular intervals and changes, such as in finance or science. Ultimately, mastering these concepts expands one's capacity to capture and manipulate mathematical patterns with greater ease.