Problem 32 - Precalculus: Mathematical for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$5,5^{c+1}, 5^{2 c+1}, 5^{3 c+1}, \ldots$$

Step-by-Step Solution

Answer

Common ratio: \(5^c\), Fifth term: \(5^{4c+1}\), \(n\) th term: \(5^{cn-c+1}\).

Step 1: Determine the Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply any term to get the subsequent term. Given the sequence \(5, 5^{c+1}, 5^{2c+1}, 5^{3c+1}, \ldots\), the second term \(5^{c+1}\) is derived from the first term by multiplying the first term by the common ratio. To find this ratio, divide the second term by the first term: \(r = \frac{5^{c+1}}{5}\). Simplifying this gives us the common ratio \(r = 5^{c}\).

Step 2: Calculate the Fifth Term

The formula for the \(n^{th}\) term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. The first term \(a_1\) is \(5\) and the common ratio \(r\) is \(5^{c}\). To find the fifth term, we set \(n = 5\) and compute: \[a_5 = 5 \cdot (5^{c})^{4} = 5 \cdot 5^{4c} = 5^{4c+1}.\] Thus, the fifth term is \(5^{4c+1}\).

Step 3: Find the nth Term

Using the formula for the \(n^{th}\) term of a geometric sequence \(a_n = a_1 \cdot r^{n-1}\), and substituting the first term \(a_1 = 5\) and the common ratio \(r = 5^{c}\), we find: \[a_n = 5 \cdot (5^{c})^{n-1} = 5 \cdot 5^{c(n-1)} = 5^{c(n-1) + 1}.\] This simplifies the \(n^{th}\) term to \(a_n = 5^{cn - c + 1}\).

Key Concepts

Common Ratio

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding the common ratio is crucial as it dictates how the sequence progresses.
In the given sequence, \[5, 5^{c+1}, 5^{2c+1}, 5^{3c+1}, \ldots\],
the common ratio must be identified to understand how each term changes to the next. To find this ratio, divide the second term by the first:
\[r = \frac{5^{c+1}}{5} = 5^c.\]
Here, \(5^c\) is the common ratio. This means each term is multiplied by \(5^c\) to yield the next term. By recognizing this pattern, it becomes easy to predict or calculate subsequent terms.

nth Term Formula

The nth term of a geometric sequence expresses the general formula for finding any term in the sequence. This formula is vital because it links every term back to the first term without the need to compute all preceding terms.

• Formula: The formula used for the nth term of a geometric sequence is:
  \[a_n = a_1 \cdot r^{n-1},\]
  where \(a_1\) is the first term and \(r\) is the common ratio.
• Application: For our sequence, where \(a_1 = 5\) and \(r = 5^c\), substituting these values provides us:
  \[a_n = 5 \cdot (5^c)^{n-1} = 5 \cdot 5^{c(n-1)}.\]
  Simplifying further gives:
  \[a_n = 5^{cn - c + 1}.\]
  This formula enables us to calculate any term's value directly by plugging in the desired term number \(n\).

Exponential Expressions

Exponential expressions are powerful mathematical tools used to express repetitive multiplication of a number. In geometric sequences, exponential expressions provide a concise way to express complex sequences.

• Understanding Exponents: An exponent indicates how many times a base (e.g., 5 in our sequence) is multiplied by itself. For example, \(5^{3}\) means \(5 \times 5 \times 5\).
• Role in Geometric Sequences: For the given sequence \(5, 5^{c+1}, 5^{2c+1}, 5^{3c+1}, \ldots\), the exponents \(c+1, 2c+1, 3c+1\), etc., directly relate to term position and progression. Each successive term includes a base raised to increasingly higher powers.
• Exponential Simplification: Using the property \(a^m \cdot a^n = a^{m+n}\), exponential expressions can be rearranged and simplified, which is particularly useful for finding specific terms in a sequence.

Exponents thus not only compactly express each term in the sequence but also simplify calculations of higher order terms by allowing the application of multiplication and exponentiation properties.