Problem 91 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Use the formula \(a_{n}=a_{1} 3^{n-1}\) to find the seventh term of the sequence \(11,33,99,297, \ldots\)

Step-by-Step Solution

Answer

The seventh term of the sequence is \(11 * 729 = 8009\).

Step 1: Identify the first term and the common ratio

In the provided sequence, it can be seen that 11 is the first term, represented as \(a_{1}\). The ratio between any term and its preceding term appears to be 3 (since \(33/11 = 3\), \(99/33 = 3\), etc.).

Step 2: Use the geometric sequence formula

Plugging the values into the formula \(a_{n}=a_{1} * 3^{n-1}\) will give the seventh term. With \(n = 7\), \(a_{1} = 11\), and ratio = 3, this becomes \(a_{7}=11 * 3^{7-1}\).

Step 3: Simplify the term

Compute the expression to find the seventh term. Firstly, simplify \(3^{7-1}\) to \(3^6\) which equals 729. Then multiply 11 by 729.

Key Concepts

Common Ratio

In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed number. This fixed number is known as the common ratio. Identifying the common ratio is essential for solving any problem involving geometric sequences.

Here's how you can find it:

• Take two consecutive terms from the sequence.
• Divide the second term by the first to get the common ratio.

In our exercise, the sequence provided is 11, 33, 99, 297, etc. If we divide 33 by 11, we get 3. Similarly, dividing 99 by 33 also gives us 3. This consistency shows that the common ratio for this sequence is 3. An easy way to think about this is you're finding how much you're scaling each number to get the next. Understanding the concept of the common ratio simplifies our approach to complex sequence problems.

Seventh Term

The seventh term of a sequence in mathematics refers to the value that occupies the seventh position in the sequence. Calculating this accurately is key to understanding the progression of a sequence. For geometric sequences, we use a specific formula to determine any term's value based on its position.In our specific exercise, we are tasked with finding the seventh term of the sequence 11, 33, 99, 297, etc. Using the information:

• First term ( \(a_1\) ) is 11
• Common ratio (multiply factor) is 3
• The position, \(n\) , is 7 for the seventh term

Using the geometric sequence formula, we'll input these values to calculate the seventh term. We'll plug them into our equation to get the exact value.

Sequence Formula

A sequence formula in mathematics is a generalized way to find any term in a sequence without listing all of them. For geometric sequences, we use the formula \(a_n = a_1 \, r^{n-1}\).This formula helps us efficiently determine any term in the sequence.Let's break this formula down:

• \(a_n\) is the term we are trying to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio (how much each term is multiplied to get the next term).
• \(n\) is the term number we want to find.

By substituting the known values: \(a_1 = 11\), \(r = 3\), \(n = 7\), into the formula, we can find the seventh term: \(a_7 = 11 \, \times \, 3^{7-1}\). Calculating this further will give the result. Understanding and using this formula helps in solving problems involving geometric sequences promptly.