Problem 12 - University Calculus: Early Transcendentals

Question

Express the sums in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$1+4+9+16$$

Step-by-Step Solution

Answer

\( \sum_{i=1}^{4} i^2 \)

Step 1: Identify the Sequence Form

Observe the given numbers: 1, 4, 9, and 16. Notice that these are perfect squares of the natural numbers starting from 1. So, the sequence can be represented as: \( 1^2, 2^2, 3^2, 4^2 \).

Step 2: Write the General Form

The nth term of a perfect square sequence can be represented as \( n^2 \). In our case, the numbers are \( 1^2, 2^2, 3^2, 4^2 \). Thus, the general term is \( i^2 \) where \( i \) starts from 1 and goes to 4.

Step 3: Express as Summation

The sequence \( 1^2 + 2^2 + 3^2 + 4^2 \) can be expressed in sigma notation as: \[ \sum_{i=1}^{4} i^2 \]. Here, the index of summation \( i \) starts at 1 and ends at 4.

Key Concepts

Perfect Squares

Perfect squares are numbers that are the result of a number being multiplied by itself. This means if you multiply a number by itself, you get a perfect square. For instance, multiplying 2 by 2 gives you 4, which is a perfect square.
In terms of a sequence, numbers like 1, 4, 9, and 16 are perfect squares corresponding to the numbers 1, 2, 3, and 4 respectively.
It's helpful to recognize these patterns in sequences as they are frequently encountered in mathematical problems.

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a common difference between consecutive terms.
This type of sequence is not directly related to perfect squares, but understanding it helps in recognizing different types of patterns in sequences.
For example, in an arithmetic sequence such as 2, 4, 6, 8, each number increases by 2, which is the common difference.

• A key characteristic is that the difference between any two successive terms remains constant.
• Recognizing this can aid in solving problems that involve sequences of various types.

Although perfect squares don’t follow an arithmetic pattern, understanding arithmetic sequences strengthens overall comprehension of number patterns.

Summation Index

In sigma notation, the summation index is a variable that keeps track of the term position in a sequence.
In our example, the summation index is denoted as \( i \).
It indicates the starting and ending points for the sequence we are summing up.

• The index \( i \) starts with an initial value, such as 1, and ends with a maximum value, such as 4 in this case.
• This index is crucial in sigma notation as it defines the range over which summation occurs.

The key idea is to have a variable, like \( i \), that allows systematic addition of sequence elements according to a defined range.

Sequence Representation

Sequence representation involves expressing the terms of a sequence in a recognizable pattern or formula.
The representation is essential for understanding and manipulating sequences mathematically.
In our context, the sequence of perfect squares can be represented as \( i^2 \).
Here, \( i^2 \) represents the square of each term's position in the sequence, such as \( 1^2, 2^2, 3^2, 4^2 \).

• This representation clearly indicates how each term of the sequence is derived.
• Formulas like \( i^2 \) make it easier to analyze and calculate sequences using tools like sigma notation.

A clear sequence representation lays the groundwork for solving more complex mathematical problems involving sequences.