Problem 8 - Algebra and Trigonometry

Question

Write the first five terms of each geometric sequence. $$ a_{n}=-6 a_{n-1}, \quad a_{1}=-2 $$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are -2, 12, -72, 432, -2592

Step 1: Understand the formula

The given sequence is \(a_{n}=-6 a_{n-1}\). This means that term \(a_{n}\) is obtained by multiplying the previous term \(a_{n-1}\) by -6

Step 2: Find the first term

The first term of the sequence is given as -2, so \(a_{1}=-2\)

Step 3: Find the second term

Apply the formula to get the second term, \(a_{2}=-6*a_{1}=-6*(-2)=12\)

Step 4: Find the third term

Similarly, the third term will be \(a_{3}=-6*a_{2}=-6*12=-72\)

Step 5: Find the fourth term

The fourth term will be \(a_{4}=-6*a_{3}=-6*(-72)=432\)

Step 6: Find the fifth term

The fifth term will be \(a_{5}=-6*a_{4}=-6*432=-2592\)

Key Concepts

Understanding the First Term

In a geometric sequence, the "first term" is the starting point for the entire sequence. It's like the foundation of a building; everything else is built around it.
In the provided exercise, the first term is given as \(a_1 = -2\). This means our sequence begins with the number -2.
Knowing the first term is crucial because it directly influences all other terms in the sequence.

• It's the term we use to calculate the next term in the sequence by applying the formula.
• It helps in finding a pattern that the sequence follows.

Always ensure you have the correct first term before proceeding with further calculations, as any mistake here affects the entire sequence!

Exploring the Recursive Formula

A recursive formula in a geometric sequence tells us how to calculate each term based on the previous one. It's like a recipe guiding us step-by-step through the process of baking a cake, ensuring each ingredient goes in the right order.
For the given sequence, the recursive formula is \(a_n = -6 a_{n-1}\).

• This formula indicates the rule the sequence follows: multiply the previous term \(a_{n-1}\) by -6 to obtain the next term \(a_n\).
• Every term after the first is connected to the previous term, creating a chain of relationships that defines the sequence's structure.

Understanding recursive formulas is essential for predicting future terms of a sequence and unraveling the sequence's inherent pattern.

Executing Term Calculation

Term calculation in a geometric sequence involves using the first term and recursive formula to find successive terms. Let's break down how each term is derived using our exercise as an example.

• First Term (already known): \(a_1 = -2\)
• Second Term: To find \(a_2\), apply the formula: \(a_2 = -6 \times a_1 = -6 \times (-2) = 12\)
• Third Term: Similarly, \(a_3 = -6 \times a_2 = -6 \times 12 = -72\)
• Fourth Term: \(a_4 = -6 \times a_3 = -6 \times (-72) = 432\)
• Fifth Term: Finally, \(a_5 = -6 \times a_4 = -6 \times 432 = -2592\)

By ensuring each step is clear and precise, you can confidently calculate further terms in the sequence. Remember, practice makes perfect when it comes to term calculation, so don't hesitate to try calculating different sequences to strengthen your understanding!