Problem 40 - 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 whose \(n\)th term is given. \(1^{2}, 2^{2}, 3^{2}, 4^{2}, \ldots\)

Step-by-Step Solution

Answer

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

Step 1: Understand the sequence

The sequence is given as the squares of natural numbers: \(1^2, 2^2, 3^2, 4^2, \ldots\). This implies the general term \(T_n = n^2\).

Step 2: Calculate the first term

Calculate the first term of the partial sum sequence, \(S_1 = T_1 = 1^2 = 1\).

Step 3: Compute the second partial sum

Add the second term to the first: \(S_2 = S_1 + T_2 = 1 + 2^2 = 1 + 4 = 5\).

Step 4: Compute the third partial sum

Add the third term to the sum of the first two terms: \(S_3 = S_2 + T_3 = 5 + 3^2 = 5 + 9 = 14\).

Step 5: Calculate the fourth partial sum

Add the fourth term to the sum of the first three terms: \(S_4 = S_3 + T_4 = 14 + 4^2 = 14 + 16 = 30\).

Step 6: Calculate the fifth partial sum

Add the fifth term to the sum of the first four terms: \(S_5 = S_4 + T_5 = 30 + 5^2 = 30 + 25 = 55\).

Step 7: Calculate the sixth partial sum

Add the sixth term to the sum of the first five terms: \(S_6 = S_5 + T_6 = 55 + 6^2 = 55 + 36 = 91\).

Key Concepts

Sequence of Squares

A sequence of squares is a list where each term is the square of a natural number. For example, the sequence starts as:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)

This sequence continues infinitely. Squaring a number means multiplying it by itself. So, for any natural number \(n\), its square is \(n \times n = n^2\).
Using squares helps in understanding geometric shapes and properties like area calculations.

Natural Numbers

Natural numbers are the numbers we use for counting or labeling. They start from 1 and go on indefinitely as 1, 2, 3, 4, 5, and so on. These numbers are positive and do not include zero in many cases, though sometimes zero is also considered natural.
Natural numbers have lots of interesting properties. They are used in everyday counting and ordering. Also, operations on natural numbers, like addition or multiplication, lead to more natural numbers.
In sequences, we generally use natural numbers to define the position of terms in the sequence, helping identify patterns and relationships.

Progressions

Progressions are sequences where each new term is derived from its position using a specific rule or formula. There are different types of progressions, like arithmetic and geometric.
In an arithmetic progression, the difference between consecutive terms is constant. In our context, the sequence of squares is not an arithmetic progression because the difference between consecutive terms varies. For example, between

• \(1^2\) and \(2^2\), the difference is 3
• between \(2^2\) and \(3^2\), it's 5, and so on.

However, the position of each term in a sequence of squares is indicated by \(n\), giving us an organized progression of calculations.

Series Sum Computation

Calculating the sum of a series involves adding consecutive terms of a sequence to generate a partial sum. For the sequence of squares, this means adding terms like \(1^2\), \(2^2\), and so on, one by one.
A partial sum \(S_n\) refers to the sum of the first \(n\) terms of the sequence. As seen in the given solution:

• \(S_1 = 1\)
• \(S_2 = 5\)
• \(S_3 = 14\)

These sums tell us how much we've added up to a specific point. Computing partial sums step-by-step helps in understanding and visualizing series.
This concept is essential not just for sequences of squares, but for any series, as sums are foundational in calculus, probability, and many areas of mathematics.