Problem 74 - Algebra and Trigonometry

Question

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{8-\frac{3}{4} n\right\\} $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \( -\frac{3}{4} \). The sum of the first 50 terms is \( -518.75 \).

Step 1: Identify the Sequence Type

Examine the given sequence: \ \[ a_n = 8 - \frac{3}{4}n \]. This is a linear function of \( n \), meaning it's of the form \( a_n = a + bn \), which indicates an arithmetic sequence.

Step 2: Determine the Common Difference

Calculate the difference between successive terms. \ \[ d = a_{n+1} - a_n \ = \left( 8 - \frac{3}{4}(n+1) \right ) - \left( 8 - \frac{3}{4}n \right) \ = 8 - \frac{3}{4}n - \frac{3}{4} + \frac{3}{4}n \ = -\frac{3}{4} \] \ Therefore, the common difference is \( d = -\frac{3}{4} \).

Step 3: Find the Sum of the First 50 Terms

Use the sum formula for an arithmetic sequence, \[ S_n = \frac{n}{2} (2a + (n-1)d) \], where \( n = 50 \), \( a = 8 \), and \( d = -\frac{3}{4} \). \ Calculate: \ \[ S_{50} = \frac{50}{2} (2 \cdot 8 + (50-1)(-\frac{3}{4})) \ = 25 (16 + 49(-\frac{3}{4})) \ = 25 (16 - 36.75) \ = 25 (-20.75) \ = -518.75 \]

Key Concepts

common difference

When dealing with arithmetic sequences, one of the core concepts is the 'common difference.' This is a constant value that you add to each term to get to the next term in the sequence. It is denoted by the letter \( d \). To find the common difference, you can subtract any term from the subsequent term in the sequence. For example, in our given sequence \( a_n = 8 - \frac{3}{4} n \), you can find the common difference by computing:

\( d = a_{n+1} - a_n \).

Following the solution, we calculated:

• \( d = \left( 8 - \frac{3}{4}(n+1)\right ) - \left( 8 - \frac{3}{4} n \right) = -\frac{3}{4} \)

This tells us that each term in the sequence is \( -\frac{3}{4} \) less than the previous term. Now that we have the common difference, it makes the sequence arithmetic.

sum of arithmetic sequence

Another important concept is finding the sum of a certain number of terms in an arithmetic sequence. Known as the 'sum of arithmetic sequence,' it can be calculated using the formula:

\( S_n = \frac{n}{2} (2a + (n-1)d ) \)

Here, \( S_n \) represents the sum of the first \( n \) terms, \( a \) is the first term, and \( d \) is the common difference. For our sequence, the first term \( a \) is 8, the common difference \( d \) is \( -\frac{3}{4} \), and we want to find the sum of the first 50 terms (\( n = 50 \)).

Plugging in the values, we get:

• \( S_{50} = \frac{50}{2} (2 \times 8 + (50-1)(-\frac{3}{4})) \)
• \( S_{50} = 25 (16 + 49(-\frac{3}{4})) \)
• \( S_{50} = 25 (16 - 36.75) \)
• \( S_{50} = 25 (-20.75) \)
• \( S_{50} = -518.75 \)

This calculation finds the sum of the first 50 terms to be -518.75.

Understanding this concept is crucial as it often appears in various mathematical problems and applications.

linear function

An arithmetic sequence can often be described as a 'linear function.' This means that each term increases or decreases by a constant rate, which we call the common difference. The general form of a linear function in the context of sequences can be written as:

\( a_n = a + (n-1)d \)

For the given sequence, instead of starting with the first term \( a_1 \) and adding \( d \) repeatedly, we noticed it was easier to represent it directly using a linear function of \( n \). Hence it is given as:
\( a_n = 8 - \frac{3}{4} n \).

This linear representation simplifies understanding the sequence's structure. Here, the coefficient of \( n \) (\( -\frac{3}{4} \) ) directly corresponds to the common difference, while 8 is the term when \( n \) is 0. It’s important to interpret this structure as it provides a clear framework for identifying the behavior of the sequence over large intervals.

In summary, treating arithmetic sequences as linear functions helps in visualizing and computing essential elements such as the common difference and sums.