Problem 124 - 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 sum is expanded as \(1 + 2 + 4 + 8\).

Step 1: Identify the given summation

We are given the summation expression \( \sum_{n=0}^{3} a_{n} \). This means we will sum the terms \(a_0, a_1, a_2,\) and \(a_3\).

Step 2: Calculate the first term

We start from \(n=0\), where \(a_0 = 1\).

Step 3: Use the recursive formula to find subsequent terms

We use the recursive formula \(a_{n+1} = 2a_n\) to find each term up to \(a_3\).

Step 4: Calculate \(a_1\)

Using the previous term \(a_0 = 1\), we apply the formula: \(a_1 = 2 \times a_0 = 2 \times 1 = 2\).

Step 5: Calculate \(a_2\)

Using \(a_1 = 2\), we apply the formula: \(a_2 = 2 \times a_1 = 2 \times 2 = 4\).

Step 6: Calculate \(a_3\)

Using \(a_2 = 4\), we apply the formula: \(a_3 = 2 \times a_2 = 2 \times 4 = 8\).

Step 7: Write the sum in expanded form

The terms \(a_0\), \(a_1\), \(a_2\), and \(a_3\) are 1, 2, 4, and 8 respectively. Therefore, the expanded form of the sum is \(1 + 2 + 4 + 8\).

Key Concepts

Recursive Formula

In calculus, a recursive formula is a key concept that helps define a sequence in terms of its preceding terms. In simpler terms, instead of giving you each number in a series outright, it provides a way to calculate the next number based on the previous one.

For our exercise, the recursive formula is given as \(a_{n+1} = 2a_n\). This tells us how to find any term of the series if we know the term before it:

• The formula shows how to "double" the previous term to get the next one.

Let's break it down:

• Start with \(a_0 = 1\).
• For each next term, multiply the previous term by 2.

This recursive rule helps us efficiently calculate each term without having to compute it from scratch every time. By using this formula, we can quickly find subsequent terms like \(a_1 = 2\), \(a_2 = 4\), and \(a_3 = 8\). It's like building a sequence, step by step!

Expanded Form

When we talk about writing a summation in expanded form, we're essentially unfolding the sum so we can clearly see all the components. It's like opening a box to see what's inside.

An expanded form writes out each term in the series one by one, rather than just providing a compact formula like \( \sum \). For our exercise:

• We have to find the sum \( \sum_{n=0}^{3} a_{n} \).
• We calculated that \( a_0 = 1 \), \( a_1 = 2 \), \( a_2 = 4 \), and \( a_3 = 8 \).

To express this in expanded form, we write: \[1 + 2 + 4 + 8\]This gives a clear picture of what the series adds up to. Expanded forms are particularly useful for understanding computations at a glance, as well as verifying calculations and checking the logic of recursive sequences.

Series

A series in calculus refers to the sum of the terms of a sequence. The term "series" might sound grand, but it's just about adding certain numbers from a sequence together.

In the context of our current exercise, when we calculate the series \(\sum_{n=0}^{3} a_{n}\), we're summing up the terms from \(n=0\) to \(n=3\), which we have already figured out are 1, 2, 4, and 8. The series therefore equals:

• \(1 + 2 + 4 + 8\)

Series help us describe quantities that grow or shrink according to a predictable pattern. They're applicable in various fields, from finance when calculating compound interest to physics in describing wave patterns.
Understanding series can make solving complex problems easier, as they provide a structured way to think about adding sequences of numbers. They can sometimes even lead to formulas that describe infinitely many terms, which is a fascinating topic in advanced calculus.