Problem 34 - Algebra and Trigonometry

Question

\(33-36\) . 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: Identify the Sequence

The sequence given is the sequence of perfect squares: \(1^2, 2^2, 3^2, 4^2, \ldots\). This means the terms of the sequence are 1, 4, 9, 16, and so on.

Step 2: Define Partial Sums

A partial sum \(S_n\) of a sequence is the sum of the first \(n\) terms of that sequence. We need to find the first six partial sums: \(S_1, S_2, S_3, S_4, S_5, S_6\).

Step 3: Calculate \(S_1\)

\(S_1\) is the sum of the first term of the sequence. Therefore, \(S_1 = 1^2 = 1\).

Step 4: Calculate \(S_2\)

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

Step 5: Calculate \(S_3\)

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

Step 6: Calculate \(S_4\)

\(S_4\) 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 \(S_5\)

\(S_5\) 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 \(S_6\)

\(S_6\) 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

sequence of perfect squares

The phrase "sequence of perfect squares" refers to a list of numbers, where each number is the square of a natural number. A natural number is any positive, whole number starting from 1, like 1, 2, 3, and so on.
For example:

• When squaring the number 1, we get 1² which equals 1.
• Likewise, 2 squared (2²) gives 4.
• Continue this with 3, we get 3² equals 9.

This pattern continues indefinitely, meaning there's an endless amount of numbers that can be created using this pattern. Understanding the pattern of perfect squares can help when you're tasked with finding sums of sequences, as it provides a consistent and predictable order of numbers to work from.

sum of sequence

Finding the sum of a sequence involves adding up the numbers within that sequence. In our case with the sequence of perfect squares, we are adding squares of consecutive natural numbers.
To calculate these, consider taking each square number:

• Start with the first number, such as 1 (from 1²).
• Add that to the next number, 4 (from 2²).
• Continue this process by adding the result to 9 (from 3²), and so forth.

This addition process provides what is known as a partial sum when you add only part of the sequence. It starts simple and helps us to digest larger sums later on by working through them step by step.

partial sum calculation

The concept of partial sums is integral when working with sequences because it allows us to systematically compute the sum of terms incrementally in a sequence without having to handle all numbers at once. For example, calculating the partial sums of our perfect squares sequence involves the following process:

• First, identify the initial term and note it as your first partial sum: \(S_1 = 1^2 = 1\).
• Next, add the subsequent term from the sequence to calculate the next partial sum: \(S_2 = 1^2 + 2^2 = 5\).
• Continue this by adding the next term for \(S_3 = 1^2 + 2^2 + 3^2 = 14\), and so on.

By breaking it down this way, students can follow a natural progression of adding one square at a time until they reach their targeted partial sum, making the overall process less intimidating and easier to manage.