Problem 30 - College Algebra

Question

Write the first five terms of the sequence. $$a_{1}=-30, a_{n}=\left(2+a_{n-1}\right)\left(\frac{1}{2}\right)^{n}$$

Step-by-Step Solution

Answer

The first five terms are \(-30, -7, -\frac{5}{8}, \frac{11}{128}, \frac{267}{4096}\).

Step 1: Write down the first term of the sequence

The first term of the sequence is provided in the problem. It is \( a_1 = -30 \).

Step 2: Use the recursion formula for the second term

To find the second term \( a_2 \), substitute \( n = 2 \) and \( a_1 = -30 \) into the formula: \[ a_2 = ig(2 + a_1ig) igg(\frac{1}{2}\bigg)^2 = ig(2 - 30ig) igg(\frac{1}{4}\bigg) = -28 \times \frac{1}{4} = -7 \] Thus, the second term is \( a_2 = -7 \).

Step 3: Calculate the third term using the recursion formula

For the third term \( a_3 \), substitute \( n = 3 \) and \( a_2 = -7 \) into the formula: \[ a_3 = ig(2 + a_2ig) igg(\frac{1}{2}\bigg)^3 = ig(2 - 7ig) igg(\frac{1}{8}\bigg) = -5 \times \frac{1}{8} = -\frac{5}{8} \] Thus, the third term is \( a_3 = -\frac{5}{8} \).

Step 4: Find the fourth term using the formula

For the fourth term \( a_4 \), substitute \( n = 4 \) and \( a_3 = -\frac{5}{8} \) into the formula: \[ a_4 = igg(2 + \frac{-5}{8} \bigg) \bigg(\frac{1}{2}\bigg)^4 = \bigg(\frac{11}{8}\bigg) \bigg(\frac{1}{16}\bigg) = \frac{11}{128} \] Thus, the fourth term is \( a_4 = \frac{11}{128} \).

Step 5: Calculate the fifth term using the formula

For the fifth term \( a_5 \), substitute \( n = 5 \) and \( a_4 = \frac{11}{128} \) into the formula: \[ a_5 = igg(2 + \frac{11}{128} \bigg) \bigg(\frac{1}{2}\bigg)^5 = \bigg(\frac{267}{128}\bigg) \bigg(\frac{1}{32}\bigg) = \frac{267}{4096} \] Thus, the fifth term is \( a_5 = \frac{267}{4096} \).

Key Concepts

Sequence Terms

A sequence is essentially a list of numbers in a specific order where each term can follow a distinct pattern. Here, the given sequence begins with a number known as the first term. In this problem, the first term is provided directly: \( a_1 = -30 \). Knowing the initial term is crucial because, in recursive sequences like this one, each new term is derived from the previous one using a set formula. By understanding the significance of the initial terms, students can accurately generate other terms by using the recursive relation.

Recursive Sequence

A recursive sequence defines each term using the previous terms in the sequence. It's like building a tower where each level depends on the one directly below it. In this exercise, the sequence is specified recursively with the formula:

• \( a_n = \left(2 + a_{n-1}\right)\left(\frac{1}{2}\right)^n \)

This formula tells us that to find \( a_n \), the \( n^{th} \) term, you need to know \( a_{n-1} \), the \( (n-1)^{th} \) term. With the first term known, you apply the formula step-by-step to generate subsequent terms. This requires plugging the preceding term into the formula along with \( n \). The sequence undergoes significant changes with each calculation, showing how recursive relations can create rapidly evolving sequences.

Arithmetic Calculations

Arithmetic calculations are at the heart of finding each term in a recursive sequence. Each step involves:

• Substituting the previous term.
• Applying given constants and multipliers, such as \( 2 \) and \( \frac{1}{2} \) raised to the power of \( n \).
• Executing operations like addition and multiplication. For instance, for the term \( a_2 \), computations involve: \( \left(2 - 30\right) \times \frac{1}{4} = -28 \times \frac{1}{4} = -7 \).

Understanding each operation is crucial to correctly deriving the term value. These calculations become more complex with further terms but follow the same logical steps. Employing these mathematical practices allows the accurate progression from one term to the next within a recursive sequence.