Problem 56 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We have an arithmetic sequence where the second term \(a_2 = 8\) and the fifth term \(a_5 = 9.5\). We need to find the partial sum \(S_{15}\) of the first 15 terms of this sequence.

Step 2: Finding the Common Difference

Given the sequence is arithmetic, the common difference \(d\) is the same between each term. The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\). Setting up the equations for \(a_2\) and \(a_5\), we have:1. \(a_2 = a_1 + d = 8\)2. \(a_5 = a_1 + 4d = 9.5\)Subtract the first equation from the second: \[a_5 - a_2 = (a_1 + 4d) - (a_1 + d) = 9.5 - 8\]This simplifies to \(3d = 1.5\), hence \(d = 0.5\).

Step 3: Finding the First Term

Using \(a_2 = a_1 + d\) and \(d = 0.5\), we substitute to find \(a_1\):\[a_1 + 0.5 = 8\]Solving for \(a_1\) gives \(a_1 = 7.5\).

Step 4: Calculating the 15th Term

Using the formula \(a_n = a_1 + (n-1)d\), we find \(a_{15}\) as:\[a_{15} = 7.5 + (15-1) imes 0.5 = 7.5 + 7 = 14.5\]

Step 5: Calculating the Partial Sum

The formula for the sum of the first \(n\) terms of an arithmetic sequence is given by:\[S_n = \frac{n}{2} (a_1 + a_n)\]Substituting \(n = 15\), \(a_1 = 7.5\), and \(a_{15} = 14.5\), we calculate:\[S_{15} = \frac{15}{2} (7.5 + 14.5) = \frac{15}{2} \times 22 = 15 \times 11 = 165\]

Key Concepts

Common Difference

In an arithmetic sequence, the common difference, often denoted as \(d\), is what sets it apart from other sequences. This number is consistent, and it represents the difference between any two successive terms. To find the common difference, use the formula \(d = a_{n} - a_{(n-1)}\), where \(a_{n}\) and \(a_{(n-1)}\) are any two consecutive terms in the sequence.

In our problem, we know the second term \(a_2 = 8\) and the fifth term \(a_5 = 9.5\). By establishing the equations using the general term formula of an arithmetic sequence, \(a_n = a_1 + (n-1)d\), we can solve for \(d\) by setting up the differences:

• \(a_2 = a_1 + d\)
• \(a_5 = a_1 + 4d\)

Subtracting these equations gives us the result \(3d = 1.5\), hence \(d = 0.5\). This value indicates that, for every step to the next term in this sequence, we add 0.5.

Partial Sum

A partial sum in an arithmetic sequence is the sum of a specified number of terms from the beginning of the sequence. The formula to find the partial sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by: \[S_n = \frac{n}{2} (a_1 + a_n)\]This formula balances the simplicity of the arithmetic mean with the range given by the first and nth terms.

For our example, we need the sum of the first 15 terms. We already discovered that \(a_1 = 7.5\) and \(a_{15} = 14.5\). Plug these back into the formula:

• \(S_{15} = \frac{15}{2} (7.5 + 14.5)\)
• After computing, \(S_{15} = \frac{15}{2} \times 22\)
• This simplifies to \(15 \times 11 = 165\)

Thus, the sum of the first 15 terms is 165.

nth Term

The \(n\)-th term of an arithmetic sequence is a vital concept because it allows you to find any term without listing all previous ones. The formula for the \(n\)-th term is:\[a_n = a_1 + (n-1)d\]
This formula requires the first term \(a_1\), the position of the term \(n\), and the common difference \(d\). It is a straightforward yet powerful tool for understanding patterns in sequences.

In solving our problem, we specifically need the fifteenth term \(a_{15}\). Utilizing the previously calculated values \(a_1 = 7.5\) and \(d = 0.5\), the calculation becomes:

• \(a_{15} = 7.5 + (15-1) \times 0.5\)
• Simplifying, we find \(a_{15} = 7.5 + 7 = 14.5\)

Hence, the fifteenth term is 14.5, confirming the progression of our arithmetic sequence.