Problem 34 - Precalculus Mathematics 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 partial sums are: \(S_1=1\), \(S_2=5\), \(S_3=14\), \(S_4=30\), \(S_5=55\), \(S_6=91\).

Step 1: Identify the Sequence

The sequence is given as the squares of positive integers, which means the sequence is \(1^2, 2^2, 3^2, 4^2, \ldots\). This sequence can be written as \(1, 4, 9, 16, \ldots\).

Step 2: Calculate the First Partial Sum, \(S_1\)

The first partial sum, \(S_1\), is the sum of the first term of the sequence. Therefore, \(S_1 = 1\).

Step 3: Calculate the Second Partial Sum, \(S_2\)

Add the first two terms of the sequence. \(S_2 = 1 + 4 = 5\).

Step 4: Calculate the Third Partial Sum, \(S_3\)

Add the first three terms of the sequence. \(S_3 = 1 + 4 + 9 = 14\).

Step 5: Calculate the Fourth Partial Sum, \(S_4\)

Add the first four terms of the sequence. \(S_4 = 1 + 4 + 9 + 16 = 30\).

Step 6: Calculate the Fifth Partial Sum, \(S_5\)

Add the first five terms of the sequence. \(S_5 = 1 + 4 + 9 + 16 + 25 = 55\).

Step 7: Calculate the Sixth Partial Sum, \(S_6\)

Add the first six terms of the sequence. \(S_6 = 1 + 4 + 9 + 16 + 25 + 36 = 91\).

Key Concepts

Sequences

In mathematics, a sequence is a set of numbers arranged in a precise order, following a particular rule. It's like a list of numbers that go on one after another, each entry referred to as a term. Sequences are fundamental because they allow us to analyze patterns and predict future terms.
In the given exercise, our sequence consists of squares of positive integers such as \(1^2, 2^2, 3^2, 4^2, \ldots\). These can be simplified as \(1, 4, 9, 16, \ldots\). Each term in this sequence follows a rule: \(n^2\), where \(n\) is the position in the sequence. By understanding this generating rule, it becomes easier to determine any term in the sequence.

Sums of Squares

The sums of squares concept involves adding the squares of numbers to get a total. This is a common task in mathematics, often appearing in series and statistical formulas.
When asked to find partial sums, each 'sum' involves adding sequential terms of our original sequence. For example, the first term is \(1^2\), or 1. The next sum adds the square of \(2\), making it \(1 + 4 = 5\). Continuing this way, you can observe that the pattern of adding squares continues - making the sequence of partial sums (\(S_1, S_2, S_3\), and so on) grow relatively quickly compared to just adding the integers themselves.

Series

A series is like a sequence, but instead of listing numbers, you sum them. When you have a sequence, and you sum its terms, you create something known as a series. In the context of this exercise, we are dealing with a specific type of series: the sum of the first \(n\) squares.
Every time we add the next term in the sequence to the accumulated sum of previous terms, we progress through what's known as a partial sum of the series. Partial sums give us an understanding of how a series behaves at different levels. In mathematics, partial sums can sometimes help us approximate the sum of all terms of an infinite series, though this is not always possible. Here, they specifically show how the sums grow due to the increasing nature of squared numbers.