Problem 80 - Beginning and Intermediate Algebra

Question

A child builds a tower with blocks so that the bottom row contains 9 blocks and the top row contains 1 block. If he uses 45 blocks, how many rows are in the tower?

Step-by-Step Solution

Answer

The tower has 9 rows in total.

Step 1: Identify the arithmetic sequence

The given tower has its bottom row with 9 blocks and decreasing by 1 block until the top row contains 1 block only. This forms an arithmetic sequence: 9, 8, 7, ... , 2, 1.

Step 2: Use the formula for the sum of the arithmetic series

To find the sum of the arithmetic series, we use the formula: \[S_n = \frac{n}{2} (a_1 + a_n)\], where \(S_n\) is the sum of the series, \(n\) is the number of terms (rows) in the series, \(a_1\) is the first term (9), and \(a_n\) is the last term (1). We are given that the total number of blocks is 45, so we can set \(S_n = 45\).

Step 3: Solve for the number of rows in the series

Now, we have: \[45 = \frac{n}{2} (9 + 1)\] We can simplify and solve for \(n\): \[45 = 5n\] \[n = 9\] So, the tower has 9 rows in total.

Key Concepts

Arithmetic Series

An arithmetic series is formed by adding the terms of an arithmetic sequence. In this context, each row of the tower is part of the sequence that decreases by 1 block per row, starting from 9 at the bottom. This sequence is represented as 9, 8, 7,..., up to 1 block at the top, making it an arithmetic sequence with a constant difference of -1. The arithmetic series then is the sum of these sequence terms. Understanding this concept is crucial, as it allows us to determine the total number of blocks used in the tower by summing the number of blocks in each row, which forms the arithmetic series. Notice that the sum follows the sequence pattern closely, a critical observation for resolving many mathematical problems related to sequences.

Sum of a Series

The sum of an arithmetic series can be found using the formula:

• \[S_n = \frac{n}{2} (a_1 + a_n)\]

This formula is specifically designed for arithmetic sequences, where \(S_n\) is the sum of the series, \(n\) is the total number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. In our tower block problem, these elements are crucial. We substitute the known values into the formula, calculating the total number of blocks used in the tower.
The sum formula effectively simplifies the process of finding total sums in arithmetic series. Rather than adding each block count individually, it provides a shortcut to quickly reach the sum of the series, which highlights its central role in arithmetic problems.

Mathematical Problem Solving

Mathematical problem solving with sequences involves breaking down the problem systematically. The first step usually entails recognizing a pattern or sequence, such as identifying the arithmetic sequence in the tower, which decreases uniformly. Once a pattern is identified, applying the appropriate formulas like the sum of the series becomes feasible. In our example, it concludes with simple algebra to solve for unknowns.
This step-by-step approach — recognizing the sequence, applying the formula, and solving — not only aids in understanding but also strengthens problem-solving skills. These skills extend beyond arithmetic problems, equipping learners to approach a variety of mathematical issues confidently.

Educational Math Problems

Educational math problems, such as the tower block problem, are an excellent way for students to engage with mathematical concepts practically. These problems encourage learners to apply theoretical knowledge to real-world-like scenarios, enhancing comprehension and retention. By working through such exercises, students develop their analytical abilities and gain a deeper understanding of math fundamentals.
Focusing on arithmetic series problems reinforces various math skills, including pattern recognition, formula use, and algebraic manipulation. These educational problems serve as a solid foundation, preparing students for more complex mathematical challenges in their academic journey.