Problem 131 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. \(1+q+q^{2}+q^{3}+q^{4}+\cdots+q^{n-1}\)

Step-by-Step Solution

Answer

The sum is expressed as \( \sum_{k=0}^{n-1} q^k \).

Step 1: Identify the Pattern

The given expression is a sum of terms where each term is a power of the variable \( q \). The terms are \( 1, q, q^2, q^3, \) and so on up to \( q^{n-1} \). The exponent increases by 1 for each subsequent term. This indicates an arithmetic progression of exponents, starting from 0 up to \( n-1 \).

Step 2: Understand Sigma Notation

Sigma notation is a way to write the sum of a sequence using the Greek letter \( \Sigma \), which stands for 'sum'. In this case, the sequence is \( q^0, q^1, q^2, \ldots, q^{n-1} \). We need to express this sequence using a general term and bounds for the summation.

Step 3: Write the General Term

The general term of the sequence is \( q^k \), where \( k \) starts from 0 and ends at \( n-1 \). This represents each term in the sequence we identified earlier.

Step 4: Define the Limits of Summation

The sigma notation needs limits for \( k \). Since the sequence starts at \( k = 0 \) and goes to \( k = n-1 \), these are our lower and upper bounds, respectively.

Step 5: Combine into Sigma Notation

Put everything together in sigma notation: \[\sum_{k=0}^{n-1} q^k\]This compactly represents the sum of the sequence from \( q^0 \) to \( q^{n-1} \).

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. However, in the given exercise, the focus is on the arithmetic progression of exponents rather than the numbers themselves.

In our sequence, the exponents are increasing by 1 each time. This is a key feature of an arithmetic progression:

• The difference, known as the common difference, is consistent for each step. Here it is 1.
• The sequence of exponents starts at 0 and goes up to \(n-1\).
• Each term can be described by the function \(k\), where \(k\) varies from 0 to \(n-1\).

This pattern helps us translate a long list into a concise formula using sigma notation. Recognizing this is the first step in writing sums succinctly.

Exponents

Exponents are a shorthand used in mathematics to represent repeated multiplication. In our sequence, each term is a power of \(q\):

• \(q^0\) equals 1, since any number to the power of 0 is 1.
• \(q^1\) is simply \(q\) itself.
• \(q^2, q^3, \) and so on represent multiplying \(q\) by itself for the given number of times.

The sequence \(1, q, q^2, q^3, \ldots, q^{n-1}\) therefore shows \(q\) being raised to increasing powers.

Understanding how exponents work is crucial, as they form the basis for expressing large and small numbers efficiently. They also make recognizing patterns easier, which is essential for translating expressions into sigma notation.

Summation Limits

In the context of sigma notation, the limits of summation define where the sequence starts and ends. For the given series, the bounds are:

• The series starts at \(k = 0\). This corresponds to the first term, which is \(q^0\).
• The series ends at \(k = n-1\), reaching up to \(q^{n-1}\).

These limits, \(0\) to \(n-1\), are written below and above the sigma (\(\Sigma\)) symbol. They tell us exactly how many terms are in the sum and how to evaluate it.

By setting these boundaries, sigma notation provides a compact way of expressing sums, making complex calculations easier to handle visually and computationally. Remember, the key is to make sure the general term reflects the actual pattern and that the limits accurately represent the start and end of the series.