Problem 32 - Precalculus: Mathematical for Calculus

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$1^{2}, 2^{2}, 3^{2}, 4^{2}, \ldots$$

Step-by-Step Solution

Answer

The first six partial sums are 1, 5, 14, 30, 55, and 91.

Step 1: Understanding the Problem

We are asked to find the first six partial sums of the sequence formed by the squares of the natural numbers: \(1^2, 2^2, 3^2, \ldots\). A partial sum \(S_n\) is the sum of the first \(n\) terms of a sequence.

Step 2: Compute Square Values

List the squares of the first few natural numbers: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), and \(6^2 = 36\). These are the terms of the sequence used in our partial sums.

Step 3: Calculate First Partial Sum \(S_1\)

The first partial sum \(S_1\) is just the first term of the sequence: \(S_1 = 1^2 = 1\).

Step 4: Calculate Second Partial Sum \(S_2\)

The second partial sum \(S_2\) is the sum of the first two terms: \(S_2 = 1^2 + 2^2 = 1 + 4 = 5\).

Step 5: Calculate Third Partial Sum \(S_3\)

The third partial sum \(S_3\) is the sum of the first three terms: \(S_3 = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14\).

Step 6: Calculate Fourth Partial Sum \(S_4\)

The fourth partial sum \(S_4\) is the sum of the first four terms: \(S_4 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30\).

Step 7: Calculate Fifth Partial Sum \(S_5\)

The fifth partial sum \(S_5\) is the sum of the first five terms: \(S_5 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55\).

Step 8: Calculate Sixth Partial Sum \(S_6\)

The sixth partial sum \(S_6\) is the sum of the first six terms: \(S_6 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91\).

Key Concepts

Sequences

In mathematics, a sequence is an ordered list of numbers that follow a particular pattern or rule. Sequences can be finite or infinite. When we say sequence in this context, it's the infinite sequence of numbers:

• 1, 4, 9, 16, 25, 36, 49, etc.

Each term in the sequence is typically denoted by an index, such as the nth term, where the index n represents the position of the term in the sequence.
Whenever you see numbers like 1, 4, 9, and so forth, following a clear rule or pattern, you are looking at a sequence. This rule is what defines the sequence. For this exercise, the rule is simple: each term is the square of a natural number \(n^2\).
Understanding sequences involves recognizing the rule or formula governing the progression from one term to the next. Once you grasp the rule, predicting future terms becomes much easier.

Squares of Natural Numbers

Natural numbers are simply the set of positive integers: 1, 2, 3, and so on. They are called 'natural' because they are the numbers you naturally count with. The square of a natural number is just that number multiplied by itself.

• For example, the square of 2 is \(2^2 = 4\).
• For 3, it is \(3^2 = 9\).

Squaring natural numbers results in a sequence of numbers that grow larger quite quickly. This is because squaring a number increases it significantly compared to merely adding or multiplying it by a smaller number.
Familiarity with squares is essential in many areas of mathematics, as they form foundational building blocks for understanding more complex concepts, such as quadratic equations and the Pythagorean theorem.
In exercises involving sequences like these, you often start by listing out the squares as a way to visualize the progression of values.

Series

A series in mathematics is the sum of the terms of a sequence. The series formed by adding the first n terms of a sequence is called a partial sum. The concept of a series is essential because it helps us understand how the sum of an infinite sequence can accumulate to a finite value.
An example based on our sequence of squares would be the partial sums:

• \(S_1 = 1^2 = 1\)
• \(S_2 = 1^2 + 2^2 = 5\)
• \(S_3 = 1^2 + 2^2 + 3^2 = 14\)

This continues as you add more squares, gradually increasing the total. Understanding series is critical for summing sequences and has applications in calculus and other advanced mathematical fields.
In practical exercises, calculating partial sums helps in exploring the behavior of the sequence's growth and offers insights into how quickly the terms are accumulating.