Problem 33 - Algebra 2

Question

Write an equation for the nth term of each geometric sequence. $$ 4,-12,36, \dots $$

Step-by-Step Solution

Answer

The nth term of the sequence is \( a_n = 4 \cdot (-3)^{n-1} \).

Step 1: Identify the First Term

The first term of the geometric sequence is given as \( a_1 = 4 \). This is the starting point of the sequence.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \(-12 \div 4 = -3\). Verify by dividing the third term by the second term: \(36 \div (-12) = -3\). Hence, the common ratio \( r = -3 \).

Step 3: Write the General Formula for the nth Term

The nth term of a geometric sequence is given by the formula \( a_n = a_1 \cdot r^{n-1} \). Substituting the known values, we get \( a_n = 4 \cdot (-3)^{n-1} \).

Step 4: Verify the Formula

To verify, compute the first few terms using the formula. For \( n=1 \), \( a_1 = 4 \cdot (-3)^{1-1} = 4 \). For \( n=2 \), \( a_2 = 4 \cdot (-3)^{2-1} = -12 \). For \( n=3 \), \( a_3 = 4 \cdot (-3)^{3-1} = 36 \). The formula is consistent with the given terms.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is fundamental in determining how the sequence progresses. It represents the constant multiplier that transitions one term to the next in the sequence. Identifying this number is crucial because it allows us to understand the structure of the sequence. To calculate the common ratio, we divide any term by the previous term. For example, let's find the common ratio in the sequence given:

• Start with the second term: \[ r = \frac{-12}{4} = -3 \]
• Verify with the next: \[ r = \frac{36}{-12} = -3 \]

Both calculations yield \( r = -3 \), confirming that the common ratio is \(-3\). This means each term is multiplied by \(-3\) to get to the next. Learning to identify the common ratio is essential for constructing the sequence further.

nth Term Formula

The nth term formula in a geometric sequence allows you to find any term without listing all the preceding ones. The formula is: \[ a_n = a_1 \cdot r^{n-1} \], where - \( a_n \) is the nth term,- \( a_1 \) is the first term, and - \( r \) is the common ratio.Let's apply this to our sequence:

• We established the first term as \( a_1 = 4 \)
• And the common ratio as \( r = -3 \)

Substitute these values into our formula: \[ a_n = 4 \cdot (-3)^{n-1} \].This gives a simple way to calculate any term in the sequence. Want to find the 5th term? Just plug \( n = 5 \) into the formula! This powerful tool can save a lot of time and effort.

Sequence Verification

After deriving the nth term formula, it's crucial to verify it with the initial terms provided. This ensures that we've calculated both the common ratio and the formula accurately.To verify, calculate the first few terms from the formula and compare them to the given sequence. Let's do this:

• For \( n=1 \): \( a_1 = 4 \cdot (-3)^{1-1} = 4 \)
• For \( n=2 \): \( a_2 = 4 \cdot (-3)^{2-1} = -12 \)
• For \( n=3 \): \( a_3 = 4 \cdot (-3)^{3-1} = 36 \)

These results match the sequence given: 4, -12, 36. Verification confirms accuracy and helps us trust our formula. Always check your work, especially when dealing with sequences.