Problem 123 - Calculus for Biology and Medicine

Question

Write each sum in expanded form. $$ \sum_{n=0}^{3} a_{n} \text { where } a_{0}=1 \text { and } a_{n+1}=2 a_{n} $$

Step-by-Step Solution

Answer

The expanded form is 1 + 2 + 4 + 8, which sums to 15.

Step 1: Understand the Summation

The expression \( \sum_{n=0}^{3} a_{n} \) means that we need to find the sum of the sequence \( a_n \) from \( n=0 \) to \( n=3 \).

Step 2: Identify the Initial Term

The problem states that \( a_0 = 1 \). This is the starting point for finding the following terms in the sequence.

Step 3: Apply the Recurrence Relation

The relation \( a_{n+1} = 2a_n \) allows us to find each subsequent term.\\( a_1 = 2a_0 = 2 \times 1 = 2 \), \\( a_2 = 2a_1 = 2 \times 2 = 4 \), \\( a_3 = 2a_2 = 2 \times 4 = 8 \).

Step 4: Write the Series in Expanded Form

Using the calculated terms, the series expands as: \( a_0 + a_1 + a_2 + a_3 = 1 + 2 + 4 + 8 \).

Step 5: Summing the Series

Add the elements in the expanded sum: \\( 1 + 2 + 4 + 8 = 15 \).

Key Concepts

Recurrence Relation

In simple words, a recurrence relation is a way to define a sequence where each term is given in relation to the previous one(s). This is like having a recipe for cooking, where the next step depends on what you did in the previous step.

• The recurrence relation given here is \( a_{n+1} = 2a_n \).
• It tells us exactly how to find the next term in the sequence if we know the current term.

Starting with your initial condition \( a_0 = 1 \), use the recurrence relation to find \( a_1, a_2 \), and \( a_3 \).

• To find \( a_1 \), set \( n = 0 \): \( a_1 = 2a_0 = 2 \times 1 = 2 \).
• Next, for \( a_2 \), with \( n = 1 \): \( a_2 = 2a_1 = 2 \times 2 = 4 \).
• Finally, \( a_3 \) with \( n = 2 \): \( a_3 = 2a_2 = 2 \times 4 = 8 \).

Notice how each step follows directly from the one before it using your recurrence recipe. You multiply the current term by 2 to get to the next one!

Series Expansion

Series expansion in this context refers to taking the terms from a sequence and writing them out as a summation. Think of it like turning a list of instructions from a cooking recipe into the actual steps you follow. In our example, the series expansion involves writing out the terms found using the recurrence relation:

• Our sequence is \( a_0, a_1, a_2, a_3 \), which are \( 1, 2, 4, \text{and} 8 \).
• The expanded series or summation is \( a_0 + a_1 + a_2 + a_3 \).
• This looks like \( 1 + 2 + 4 + 8 \).

Each term is clearly visible in the expanded form, letting you easily add them together to find the total sum of the series. Expanding a series is key in understanding how much each part contributes to the whole.

Sequence

A sequence is simply an ordered list of numbers. It's like your daily schedule, where specific items occur at set times.More formally:

• Each number in a sequence is called a "term".
• A sequence can be finite (end after a particular number of terms) or infinite (continue indefinitely).

In this exercise, you're given a sequence starting with the initial term \( a_0 = 1 \) and use a recurrence relation to find the next few.

• Our sequence is \( 1, 2, 4, 8 \), corresponding to \( n = 0, 1, 2, \text{and} 3 \).
• This sequence shows exponential growth since each term doubles the previous one.

Understanding sequences allows you to comprehend patterns and make predictions about the terms that come next. In various branches of mathematics, recognizing and analyzing sequences is fundamental, aiding both in theoretical and applied contexts.