Carrying Out the Plan

Applying the generalized solution

In the previous lesson, we considered and solved only the specific problem at hand. Now, let’s apply the generalized formula of the arithmetic series to solve both the problems that we looked at in the previous lessons.

Gauss’s problem

In this case, we have a sequence of numbers from 1 to 100. The first term of the sequence is 1, and the last term is 100. The total numbers in the sequence are 100. Now, let’s use this information to find the sum:

$$\begin{align*} \text{Sum} &= \frac{\text{100}}{2}\times(1 + 100) \\ &= 50\times 101 \\ &= 5050 \\ \end{align*}$$

The sum of numbers from 1 to 100 is 5050.

Jill’s problem

Let’s now apply this formula to Jill’s problem. In this case, we have 50 numbers in the sequence, starting from 1 and going all the way to 50. The first term of the sequence is 1, and the last term is 50. The total numbers in the sequence are 50. Now, let’s use this information to find the sum:

$$\begin{align*} \text{Sum} &= \frac{\text{50}}{2}\times(1 + 50) \\ &= 25\times 51 \\ &= 1275 \\ \end{align*}$$

So, Jill will need 1275 blocks to build their 50-layer triangular wall.

Real-world applications of arithmetic series

Arithmetic series is a versatile mathematical tool that helps model various real-world phenomena involving consistent changes or patterns over time or in a sequence. Here are a few examples:

• Computer science: Arithmetic series play an important role in analyzing the time complexity of algorithms. When evaluating algorithms, it’s crucial to understand how the number of operations or steps an algorithm takes grows with the input size. By understanding how the number of operations scales with the input size using arithmetic series, computer scientists and programmers can make informed decisions about algorithm selection, optimizing code, and predicting how algorithms will perform as the input size increases.

• Finance and economics: Calculating the total amount of money spent or earned over a period of time, like the total revenue from sales or the total expenses incurred, often involves arithmetic series. Also, when calculating the amortization of loans or mortgages, the total amount paid over time follows an arithmetic series.

• Retail and inventory management: Calculating the total sales or inventory costs over a period, especially when there’s a consistent increase or decrease, can be modeled using an arithmetic series.

These applications demonstrate how arithmetic series play a significant role in various fields, helping to model, calculate, and predict real-world phenomena involving consistent changes or patterns.