Problem 73 - 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. $$ \left\\{3-\frac{2}{3} n\right\\} $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of \( d = -\frac{2}{3} \). The sum of the first 50 terms is approximately -666.667.

Step 1 - Identify the Sequence Type

First, identify the given sequence. The general term of the sequence is given by the formula: \ \( a_n = 3 - \frac{2}{3}n \).

Step 2 - Determine if the Sequence is Arithmetic

To check if the sequence is arithmetic, find the difference between consecutive terms: \ \( a_{n+1} - a_n = \left(3 - \frac{2}{3}(n+1)\right) - \left(3 - \frac{2}{3}n\right) \). Simplify the expression: \ \( 3 - \frac{2}{3}(n+1) - 3 + \frac{2}{3}n = \frac{2}{3}n - \frac{2}{3} - \frac{2}{3}n = -\frac{2}{3} \). The common difference is constant, thus the sequence is arithmetic with a common difference \( d = -\frac{2}{3} \).

Step 3 - Find the Sum of the First 50 Terms

For an arithmetic sequence, the sum of the first \( n \) terms is given by: \ \( S_n = \frac{n}{2} \, (2a + (n-1)d) \). Here, \( a = 3 \), \( d = -\frac{2}{3} \), and \( n = 50 \). Substitute into the formula: \ \( S_{50} = \frac{50}{2} \, (2 \cdot 3 + (50-1) \cdot (-\frac{2}{3})) \). Simplify: \ \( S_{50} = 25 \, (6 - 49 \cdot \frac{2}{3}) \), \ \( S_{50} = 25 \, (6 - 32.6667) \), \ \( S_{50} = 25 \, (-26.6667) \), \ \( S_{50} = -666.667 \).

Key Concepts

Common Difference

The common difference is a key concept in identifying arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is always the same. This difference is called the common difference, denoted as \(d\). To find the common difference, simply subtract any term from the term that follows it.
In the given sequence, \( a_n = 3 - \frac{2}{3}n \), let’s calculate the difference between consecutive terms: \( a_{n+1} - a_n = \big(3 - \frac{2}{3}(n+1)\big) - \big(3 - \frac{2}{3}n\big) \).
Simplifying this:
\( 3 - \frac{2}{3}(n+1) - 3 + \frac{2}{3}n \).
We get: \( \frac{2}{3}n - \frac{2}{3} - \frac{2}{3}n \).
So, \(d = -\frac{2}{3}\).
Here, \(d = -\frac{2}{3}\) is constant, confirming that the sequence is arithmetic with a common difference of \(-\frac{2}{3}\).

Sum of Arithmetic Sequence

Finding the sum of the first \(n\) terms of an arithmetic sequence requires a specific formula: \( S_n = \frac{n}{2} \big(2a + (n-1)d\big) \).
For our sequence \( a_n = 3 - \frac{2}{3}n \), we already know:

• The first term \(a = 3\)
• The common difference \(d = -\frac{2}{3}\)
• We want the sum of the first 50 terms, so \(n = 50\)

Plugging these values into the formula:
\( S_{50} = \frac{50}{2} \big(2 \times 3 + (50-1) \times -\frac{2}{3}\big) \).
Simplify step by step:
\( S_{50} = 25 \times \big(6 + 49 \times -\frac{2}{3}\big) \)
\( S_{50} = 25 \times \big(6 - 32.6667\big) \)
\( S_{50} = 25 \times -26.6667 \)
Finally,
\( S_{50} = -666.667 \).
This result represents the sum of the first 50 terms of the sequence.

Sequence Identification

Sequence identification is the initial step in solving problems involving sequences. A sequence is a set of numbers arranged in a specific order. It's important to determine whether the sequence is arithmetic, geometric, or neither.
For our sequence \( a_n = 3 - \frac{2}{3}n \), we can identify its type by looking at the pattern of the terms.

• If the difference between consecutive terms is constant, it’s an arithmetic sequence.
• If the ratio between consecutive terms is constant, it’s a geometric sequence.
• If neither, the sequence is neither arithmetic nor geometric.

By examining the general term \( 3 - \frac{2}{3}n \), we see that the difference between any two consecutive terms is \(-\frac{2}{3}\), indicating it is an arithmetic sequence. Thus, using the formula for the common difference confirms this observation.