Problem 42 - Algebra and Trigonometry

Question

The second and fifth terms of a geometric sequence are 10 and \(1250,\) respectively. Is \(31,250\) a term of this sequence? If so, which term is it?

Step-by-Step Solution

Answer

Yes, 31,250 is the 7th term of the sequence.

Step 1: Identify the formula for the nth term of a geometric sequence

In a geometric sequence, the nth term is given by the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 2: Set up equations for the given terms

We know the second term (\( a_2 = 10 \)) and the fifth term (\( a_5 = 1250 \)). Using the geometric sequence formula, we set up two equations: \( a_2 = a_1 \cdot r = 10 \) and \( a_5 = a_1 \cdot r^4 = 1250 \).

Step 3: Solve for the common ratio (r)

By dividing the equation for \( a_5 \) by the equation for \( a_2 \), we find \( r^3 = 125 \). Taking the cube root of both sides gives \( r = 5 \).

Step 4: Solve for the first term (a1)

Substitute \( r = 5 \) back into the equation \( a_1 \cdot r = 10 \) to solve for \( a_1 \). This gives \( a_1 = 2 \).

Step 5: Determine if 31,250 is a term in the sequence

We use the formula \( a_n = a_1 \cdot r^{n-1} \) to find if 31,250 can be obtained. Set \( a_n = 31250 \), giving \( 31250 = 2 \cdot 5^{n-1} \). Simplifying this results in \( 5^{n-1} = 15625 \).

Step 6: Solve for n

Since \( 15625 = 5^6 \), it follows that \( n-1 = 6 \). Therefore, \( n = 7 \). Thus, 31,250 is the 7th term of the sequence.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a fundamental component. It governs how the terms are generated sequentially. It's the factor you multiply by to go from one term to the next. To find the common ratio, you simply divide any term by its previous term. Mathematically, the common ratio \( r \) in a geometric sequence is given by dividing the nth term by the (n-1)th term: \( r = \frac{a_n}{a_{n-1}} \). For example, if you have a sequence where the second term \( a_2 \) is 10 and the fifth term \( a_5 \) is 1250, you can set up the equations \( a_2 = a_1 \cdot r \) and \( a_5 = a_1 \cdot r^4 \). By dividing these two equations, you can eliminate \( a_1 \) and solve for the common ratio \( r \). The key steps lead to an understanding of the sequence's progression.

Defining the Term of Sequence

A term in a sequence refers to a specific element or position within that sequence. In geometric sequences, each term can be expressed via the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.In practice, if you know the common ratio and the first term, you can predict any term in the sequence. For instance, to find the fifth term of a geometric sequence where \( a_1 = 2 \) and \( r = 5 \), substitute into the formula: \( a_5 = 2 \cdot 5^4 = 1250 \). Each term follows logically from the one before it by multiplication by the common ratio. This structured setup allows for the convenient calculation of any term in the sequence, helping you understand and predict further elements beyond those initially given.

The nth Term Formula Explanation

The nth term formula in a geometric sequence serves to determine any given term based on its position. This formula is \( a_n = a_1 \cdot r^{n-1} \). Here's what each part means:

• \( a_n \) is the term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term position in the sequence.

To understand this better, solving an example is helpful. Say you start with \( a_1 = 2 \) and a common ratio \( r = 5 \). If you wish to find the seventh term \( a_7 \), substitute into the formula: \( a_7 = 2 \cdot 5^{7-1} = 31250 \). By using this formula, identifying specific terms becomes a straightforward calculation, enabling detailed insights into the structure of the sequence.