Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. $$a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are 32, 16, 8, 4, 2.

Step 1: First Term

The first term \(a_{1}\) is given as 32. So, the first term is 32.

Step 2: Second Term

Apply the recursive rule to get the second term: \(a_{2}=\frac{1}{2} a_{1}= \frac{1}{2} \times 32 = 16\). So, the second term is 16.

Step 3: Third Term

Again apply the recursive rule to get the third term: \(a_{3}=\frac{1}{2} a_{2}=\frac{1}{2} \times 16 = 8\). So, the third term is 8.

Step 4: Fourth Term

Apply the recursive rule to get the fourth term: \(a_{4}=\frac{1}{2} a_{3}=\frac{1}{2} \times 8 = 4\). So, the fourth term is 4.

Step 5: Fifth Term

For the last time apply the recursive rule to get the fifth term: \(a_{5}=\frac{1}{2} a_{4}=\frac{1}{2} \times 4 = 2\). So, the fifth term is 2.

Key Concepts

Sequence Definition

Imagine a lineup of objects where the arrangement follows a specific pattern; this is essentially what we mean by a sequence in mathematics. It's a collection of numbers that are related to each other according to some rule. This rule may be straightforward, such as adding the same number to move from one term in the sequence to the next, or it may be more complex.

To put it simply, think of each number in a sequence as an individual player on a sports team. They each have their position, which in a sequence is determined by an index, often denoted with subscripts like this: \( a_1, a_2, a_3, ... \). The position tells us the order of the players, just like the numbering on a sports team's roster.

Now, when we talk about defining a sequence recursively, it's like saying you can figure out any player's capabilities by looking at the previous player's performance. The initial player's performance is a given, which in mathematics we call the 'first term' or 'initial value,' and each subsequent player improves (or changes) based on the rule applied to the player that came before.

Arithmetic Sequences

An arithmetic sequence is one of the simplest types of sequences in math. It's like climbing stairs where each step up is the same height as the last one. In numerical terms, it means that we start with a number, known as the initial term, and then consistently add (or subtract) the same value, which we call the common difference, to get the next term in the sequence.

For instance, if you start on the fifth step (our initial term), and each step is 2 inches tall (our common difference), the height you reach with every move will be your current height plus 2 inches. Here's how that looks with numbers, given starting with 5: \(a_1 = 5, a_2 = 7, a_3 = 9, \) and so on. Each term is determined by the formula \( a_n = a_1 + (n-1) \cdot d \), where \( d \) is the common difference, and \( n \) refers to the nth term in the sequence.

Geometric Sequences

Now let's get a bit more adventurous and think of a sequence as a series of bounces on a trampoline: each bounce might be a fraction of the last, but it's always a proportional change. In mathematical terms, this is a geometric sequence, where instead of adding a constant value, we multiply by a fixed number known as the common ratio.

If your first bounce is, say, 100cm high, and each subsequent bounce is half as high as the one before, your bounces form a geometric sequence. The height of each bounce can be calculated by the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio. In this case: 100cm, 50cm, 25cm, as each bounce is \(\frac{1}{2}\) of the previous one. This is closely linked to our example from the original exercise, where we see a recursive sequence with \(a_{k+1} = \frac{1}{2}a_k\), implying that the common ratio is \(\frac{1}{2}\).