Problem 44 - Precalculus: Mathematical 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 \(S_{15}\) is 165.

Step 1: Finding the Common Difference

The general formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) imes d \). We have \( a_2 = 8 \) and \( a_5 = 9.5 \). Let's write the equations based on these terms.\[ a_2 = a_1 + (2-1) imes d = a_1 + d = 8 \] \[ a_5 = a_1 + (5-1) imes d = a_1 + 4d = 9.5 \]Now subtract the first equation from the second:\[ (a_1 + 4d) - (a_1 + d) = 9.5 - 8 \] \[ 3d = 1.5 \] \[ d = 0.5 \]

Step 2: Finding the First Term

We have the equation \( a_1 + d = 8 \) from the previous step. Since \( d = 0.5 \), substitute \( d \) into the equation to find \( a_1 \):\[ a_1 + 0.5 = 8 \]\[ a_1 = 8 - 0.5 \]\[ a_1 = 7.5 \]

Step 3: Calculating the 15th Term

We need the 15th term to find the sum of the first 15 terms. Use the formula \( a_n = a_1 + (n-1) imes d \) with \( n = 15 \):\[ a_{15} = a_1 + (15-1) imes d \]\[ a_{15} = 7.5 + 14 imes 0.5 \]\[ a_{15} = 7.5 + 7 \]\[ a_{15} = 14.5 \]

Step 4: Calculating the Partial Sum

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

Key Concepts

Partial Sum

In an arithmetic sequence, a partial sum is the sum of a specific number of terms from the beginning of the sequence.
To find this sum, you use the formula for the partial sum of an arithmetic sequence: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]Where:

• \( S_n \) is the partial sum of the first \( n \) terms,
• \( n \) is the number of terms,
• \( a_1 \) is the first term,
• \( a_n \) is the nth term.

This formula is derived from the fact that each term pairs with another term equidistant from the end, adding up to a consistent total.
In the example problem, this formula helps find the sum of the first 15 terms of the given sequence.

Common Difference

The common difference in an arithmetic sequence is crucial because it represents the constant amount each term increases or decreases from the previous term.
The formula to find the common difference \( d \) is taken from two terms in the sequence such as \( a_{n} \) and \( a_{m} \):\[ d = \frac{a_n - a_m}{n - m} \]In this context,

• \( a_2 = 8 \)
• \( a_5 = 9.5 \)

To find \( d \), set up the equations:\[ a_5 = a_2 + 3d \]By solving, you will discover that \( d = 0.5 \), which means each subsequent term is obtained by adding 0.5 to the previous term.
This common difference is foundational for both deriving other terms and calculating the partial sum.

Arithmetic Sequence Formula

The arithmetic sequence formula allows you to find any term in the sequence based on the position or index of that term.
The general form of this formula is: \[ a_n = a_1 + (n-1) \times d \]Where:

• \( a_n \) is the term you are interested in
• \( a_1 \) is the first term
• \( d \) is the common difference
• \( n \) is the term number or position in the sequence

This formula enables adjusting any term's position to find its value.
For example, in the problem, you calculate \( a_{15} = 14.5 \) to plug into the partial sum formula.
The beauty of this formula is that it handles all terms in a systematic and easy way by simply changing \( n \) in the formula.

Nth Term Calculation

Finding the nth term of an arithmetic sequence is all about substituting the position number into the arithmetic formula.
Using: \[ a_n = a_1 + (n-1) \times d \]To calculate \( a_{15} \) in our problem:

• Use \( a_1 = 7.5 \)
• And \( d = 0.5 \)
• Substitute \( n = 15 \) into the formula

Firstly calculate:\[ (n-1) \times d = 14 \times 0.5 = 7 \]Then add this result to the first term:\[ a_{15} = 7.5 + 7 = 14.5 \]This systematic approach ensures you arrive at the nth term correctly.
Knowing any term’s value empowers you to solve broader questions such as total sum or sequence behavior effortlessly.