Problem 48 - Precalculus Mathematics for Calculus

Question

Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a_{2}=8, a_{5}=9.5, n=15$$

Step-by-Step Solution

Answer

The partial sum \(S_{15}\) of the sequence is 165.

Step 1: Determine the Common Difference

The common difference of an arithmetic sequence is found by subtracting the first term of the sequence from the second term and dividing the result by the number of steps between the two terms. Here, use the formulas for the 2nd and 5th terms:\[ a_2 = a_1 + d = 8 \]\[ a_5 = a_1 + 4d = 9.5 \]Subtract the first equation from the second to find the common difference, \(d\):\[ (a_1 + 4d) - (a_1 + d) = 9.5 - 8 \]\[ 3d = 1.5 \]So, \(d = 0.5\).

Step 2: Find the First Term

Now that we have the common difference, we use \(a_2\) to find the first term, \(a_1\):\[ a_2 = a_1 + d = 8 \]Substitute \(d = 0.5\) into the equation:\[ 8 = a_1 + 0.5 \]\[ a_1 = 7.5 \].

Step 3: Write the Formula for the General Term

The formula for the general term of an arithmetic sequence is given by:\[ a_n = a_1 + (n-1)d \]Substitute \(a_1 = 7.5\) and \(d = 0.5\) into this formula:\[ a_n = 7.5 + (n - 1) imes 0.5 \].

Step 4: Use the Partial Sum Formula

The formula for the partial sum of the first \(n\) terms of an arithmetic sequence is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]First, find \(a_{15}\) using \(a_n = 7.5 + (n - 1) imes 0.5\):\[ a_{15} = 7.5 + (15 - 1) imes 0.5 \]\[ a_{15} = 7.5 + 7 = 14.5 \].Then, substitute \(n = 15\), \(a_1 = 7.5\), and \(a_{15} = 14.5\) into the sum formula:\[ S_{15} = \frac{15}{2} (7.5 + 14.5) \]\[ S_{15} = \frac{15}{2} \times 22 \]\[ S_{15} = 15 \times 11 \]\[ S_{15} = 165 \].

Key Concepts

Partial Sum

In an arithmetic sequence, a partial sum is the sum of a specified number of terms starting from the beginning of the sequence. For example, if you have the first 15 terms of a sequence, the partial sum is the total of those terms. The formula to find the partial sum, particularly useful for arithmetic sequences, is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Here, \(S_n\) is the partial sum, \(n\) is the number of terms, \(a_1\) is the first term in the sequence, and \(a_n\) is the \(n^{th}\) term of the sequence.
Let's break this down:

• Divide the number of terms by 2 to prepare for averaging.
• Add together the first term and the last term you are summing (\(a_1\) and \(a_n\)).
• Multiply the average by \(n\) to get the total for the desired number of terms.

This efficient method helps find the sum without adding each term individually, saving time and effort.

Common Difference

The common difference of an arithmetic sequence is the difference between consecutive terms, which remains constant throughout the sequence. Understanding this concept is vital to decipher how the sequence progresses.To find the common difference \(d\) in an arithmetic sequence:

• Take the difference between any two consecutive terms.
• Mathematically, use \( d = a_2 - a_1 \) for easy computation, where \(a_2\) is the second term and \(a_1\) is the first.

In scenarios where you know two non-consecutive terms, the formula modifies slightly, as shown in the problem:

• Use the equations \(a_2 = a_1 + d\) and \(a_5 = a_1 + 4d\).
• Subtract these equations to isolate the common difference, solve for \(d\).

This allows you to find how much each term increases within the sequence, providing insight into its structure and rule of progression.

General Term Formula

The general term formula is a crucial tool in understanding and generating any term in an arithmetic sequence. It allows you to compute the \(n^{th}\) term directly without listing all preceding terms.The general term formula is:\[ a_n = a_1 + (n-1) \times d \]Where:

• \(a_n\) is the \(n^{th}\) term you wish to find.
• \(a_1\) is the first term in the sequence.
• \(n\) is the position of the term you are trying to find.
• \(d\) is the common difference.

Using this formula:

• Calculate how far \(n\) is from the first term, hence \((n-1)\).
• Multiply this with the common difference \(d\), to determine how much is added to the first term \(a_1\).

This formula gives a clear and direct path to determine any term's value within the sequence, essential for understanding its entire layout and structure.