Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence. $$a_{1}=5, a_{3}=\frac{45}{4}, 8 \mathrm{th} \text { term }$$

Step-by-Step Solution

Answer

To find the 8th term, calculate the common ratio using the given first and third terms, and use it in the formula for the nth term of a geometric sequence.

Step 1: Find the common ratio

The common ratio can be found using the formula \(r = (a_{3}/a_{1})^(1/(3-1))\). Substituting the given values, we have \(r = (45/4/5)^(1/2)\). Calculate this to find the value of \(r\).

Step 2: Compute for the 8th term

Once the common ratio is found, we can find the 8th term using the formula for the nth term of a geometric sequence \(a_{n} = a_{1} \cdot r^{(n-1)}\). Substituting the given first term, our found common ratio, and our n (which is 8) into this formula, we will find \(a_{8} = 5 \cdot r^{(8-1)}\). Calculate this to find the 8th term of the geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, each term progresses by multiplying the previous term by a fixed number. This number is known as the **common ratio**. To find the common ratio when you know a couple of terms in the sequence, you can take one term and divide it by the preceding term. For example, given that the first term \(a_1\) is 5 and the third term \(a_3\) is \(\frac{45}{4}\), we can find the common ratio \(r\) using the formula:
\[r = \left( \frac{a_3}{a_1} \right)^{\frac{1}{3-1}} = \left( \frac{45/4}{5} \right)^{\frac{1}{2}}\]
Using this formula, you solve for \(r\) by breaking down the fractions and applying the exponent. This might involve calculating the value inside the parentheses first, followed by taking the square root, since the exponent is \(\frac{1}{2}\). Understanding this step is crucial to progressing further into finding terms within the sequence.

Nth Term Formula

The nth term of a geometric sequence is found by using a specific formula that incorporates the first term, the common ratio, and the position of the term in the sequence. This formula comes in handy when you're aiming to find any term in the sequence without having to list all the preceding ones. The nth term formula is expressed as:
\[a_n = a_1 \cdot r^{(n-1)}\]
Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) represents the position of the term you want to find. For example, to find the 8th term when \(a_1\) is 5 and \(r\) is calculated in the prior steps, plug in the values:

• \(n = 8\)
• First term \(a_1 = 5\)
• Common ratio \(r\)

This gives you the formula:
\[a_8 = 5 \cdot r^{7}\]
By solving this, you are applying the rules of exponents and multiplication to obtain the specific term in the sequence you’re seeking.

Mathematical Calculations

Diving into mathematical calculations is central to solving many problems related to geometric sequences. It's not just about applying formulas, but understanding the logical steps behind them. In this example, we use several mathematical principles:

• Fraction simplification: Convert fractions to simpler forms to make calculations more manageable.
• Exponentiation: After finding the common ratio \(r\), you'll raise it to the power of the term number minus one, as shown in \(r^{7}\) for the 8th term.
• Multiplicative operations: The final step involves multiplying the powered common ratio by the starting term \(a_1\) to find \(a_8\).

Practicing these steps helps solidify your understanding of geometric sequences. It’s key to work through each calculation meticulously to ensure no errors, especially as mistakes in earlier steps can compound. Remember, patience and practice are your best tools in mastering mathematical calculations.