Problem 78 - 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. $$ \left\\{\left(\frac{5}{4}\right)^{n}\right\\} $$

Step-by-Step Solution

Answer

The sequence is geometric with a common ratio of \(\frac{5}{4}\). The sum of the first 50 terms is \(-5[1-(\frac{5}{4})^{50}]\).

Step 1: Identify the type of sequence

Examine the given sequence \(\frac{5}{4}\text{,}\frac{5}{4}^{2}\text{,}\frac{5}{4}^{3},...\text{,}\frac{5}{4}^{n}\). Notice that each term is a power of \(\frac{5}{4}\). This indicates the sequence is geometric since each term is obtained by multiplying the previous term by \(\frac{5}{4}\).

Step 2: Find the common ratio

A geometric sequence has a common ratio between consecutive terms. The common ratio for this sequence can be found as follows: \(\frac{\text{second term}}{\text{first term}} = \frac{\frac{5}{4}^{2}}{\frac{5}{4}} = \frac{5}{4}\).

Step 3: Use the sum formula for a geometric series

To find the sum of the first 50 terms of a geometric sequence, use the sum formula for a geometric series:\[ S_{n} = a \frac{1-r^{n}}{1-r} \] Here, \(a\) is the first term and \(r\) is the common ratio. For the given sequence, \(a = \frac{5}{4}\) and \(r = \frac{5}{4}\).Therefore, to find the sum of the first 50 terms, we use: \[ S_{50} = \frac{5}{4} \frac{1-(\frac{5}{4})^{50}}{1-\frac{5}{4}} \]

Step 4: Simplify the sum formula

Simplify the formula as follows: \[ S_{50} = \frac{5}{4} \frac{1-(\frac{5}{4})^{50}}{1-\frac{5}{4}} = \frac{5}{4} \frac{1-(\frac{5}{4})^{50}}{-\frac{1}{4}} = -5[1-(\frac{5}{4})^{50}] \]

Key Concepts

Common Ratio

In geometric sequences, the common ratio is a key concept. It's the factor by which we multiply one term to get the next. Identifying the common ratio helps confirm the sequence type and is crucial for further calculations. For example, in our given sequence \left(\frac{5}{4}\right)^n, we find the common ratio by dividing the second term by the first term: \(\frac{(\frac{5}{4})^{2}}{\frac{5}{4}} = \frac{5}{4}\). This ratio remains constant between all terms, defining the sequence as geometric. When using the common ratio in the sum formula, replace \(r\) with this value to proceed with summation.

Sum of Geometric Series

To find the sum of the first 50 terms in a geometric series, we use a specific formula. The formula is given by: \[ S_n = a \frac{1-r^n}{1-r} \]. Here, \(a\) is the first term, and \(r\) is the common ratio. For our sequence \(\frac{5}{4} , (\frac{5}{4})^2 , (\frac{5}{4})^3 , \dots , (\frac{5}{4})^n \), both \(a\) and \(r\) are equal to \(\frac{5}{4}\). Plugging these into the formula, we get: \[ S_{50} = \frac{5}{4} \frac{1-(\frac{5}{4})^{50}}{1-\frac{5}{4}} = -5[1-(\frac{5}{4})^{50}] \]. Simplifying this can get complex, especially as \(n\) increases, but understanding each step ensures correct results. Always handle fractions carefully, and perform each operation methodically.

Identification of Sequences

Determining the type of sequence is fundamental before applying further mathematical operations. A sequence generally can be arithmetic, geometric, or neither:

• **Arithmetic Sequences** progress by adding a fixed number (common difference) to each preceding term.
• **Geometric Sequences** multiply each term by a fixed number (common ratio) to get the next.
• **Neither** sequence does not follow a strict pattern of addition or multiplication.

For our specific sequence \(\frac{5}{4} , (\frac{5}{4})^2 , (\frac{5}{4})^3 , \dots , (\frac{5}{4})^n \), recognizing the pattern where each term is a power of \(\frac{5}{4}\), informs us that it's geometric. Always start identifying sequences by checking these patterns. Compare at least two consecutive terms to spot consistent additions (arithmetic) or multiplications (geometric). This initial step solidifies the foundation for solving more complex sequence problems.