Problem 66 - Essential Calculus

Question

Building Blocks A child places \(n\) cubic building blocks in a row to form the base of a triangular design (see figure). Each successive row contains two fewer blocks than the preceding row. Find a formula for the number of blocks used in the design. (Hint: The number of building blocks in the design depends on whether \(n\) is odd or even.)

Step-by-Step Solution

Answer

The final formula for the total number of blocks in the triangular design is \(S_n = \frac{n + 1}{4}(n + 1)\) if \(n\) is odd, and \(S_n = \frac{n}{2}(n + 2)\) if \(n\) is even.

Step 1: Identify the pattern

For both even and odd \(n\), the number of blocks in successive rows from top to bottom form an arithmetic sequence. The only difference is that for even \(n\), the sequence is from \(n\) to \(2\) with a difference of \(2\), and for odd \(n\), the sequence is from \(n\) to \(1\) with a difference of \(2\).

Step 2: Apply arithmetic sequence sum formula for odd n

For odd \(n\), the sum \(S_n\) of an arithmetic sequence is given by the formula \(S_n = \frac{n}{2} (a + l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Here, \(n = \frac{n + 1}{2}\), \(a = n\), and \(l = 1\). So, \(S_n = \frac{n + 1}{4}(n + 1)\).

Step 3: Apply arithmetic sequence sum formula for even n

For even \(n\), the formula for the sum \(S_n\) of an arithmetic sequence is the same, but here \(n = \frac{n}{2}\), \(a = n\), and \(l = 2\). So, \(S_n = \frac{n}{2}(n + 2)\).

Step 4: Formulate final result

The final formula for the total number of blocks in the design is \(S_n = \frac{n + 1}{4}(n + 1)\) for odd \(n\), and \(S_n = \frac{n}{2}(n + 2)\) for even \(n\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of sequence in mathematics where each term after the first is obtained by adding a constant difference to the preceding term. This difference is called the "common difference."
Understanding arithmetic sequences is crucial because they form the basis of many patterns found in nature and structures, such as the sequence used by a child constructing a triangular design with blocks.
To identify an arithmetic sequence:

• First, observe the difference between consecutive terms. If it is constant, you are dealing with an arithmetic sequence.
• Each term can be expressed as a function of the first term and the common difference.
• The general formula for the nth term is: \(a_n = a_1 + (n - 1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference.

In the context of the original exercise, the sequence of blocks decreases by 2 from one row to the next, making it an arithmetic sequence.

Sum of an Arithmetic Sequence

The sum of an arithmetic sequence, also known as an arithmetic series, is a crucial concept when evaluating the total number of items spread across several terms.
When dealing with arithmetic sequences, the sum can be simply calculated using a well-known formula devised centuries ago.
The formula for the sum \(S_n\) of an arithmetic sequence is:

• \(S_n = \frac{n}{2} imes (a + l)\)
• Where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.

For example, for the block design described in the original problem:

• If \(n\) is even, the number of rows is calculated as \(\frac{n}{2}\).
• If \(n\) is odd, the number of rows is \(\frac{n + 1}{2}\).

By substituting the correct values of \(a\) and \(l\) based on whether \(n\) is even or odd, the total number of blocks required for the design can be calculated effectively.

Even and Odd Numbers

In mathematics, understanding even and odd numbers is essential especially when creating sequences or solving problems that depend on them, like in the given exercise.
An even number is a number divisible by 2, while an odd number leaves a remainder of 1 when divided by 2.
This distinction influences patterns and calculations involved in sequences:

• Even numbers: Can be expressed in the form of \(2k\) where \(k\) is an integer.
• Odd numbers: Can be written as \(2k + 1\).

In the building block exercise, this distinction determines where sequences start and end:

• For even \(n\), the rows start at \(n\) and decrease by 2 until they reach 2.
• For odd \(n\), rows also start at \(n\) but decrease to 1.

Recognizing these patterns helps apply the correct formulas for calculating sums based on the sequence of blocks used.