Question

Chayton decides to save money in a savings account for a vacation. He deposits 10 dollar in his savings account the 1st month. Each month thereafter, the amount he deposits is 10 dollar more than the amount he deposited the previous month. Thus, Chayton's deposit is 20 dollar the 2nd month, 30 dollar the 3rd month, and so on. He makes his final deposit of 360 dollar the 36th month. What is the total amount of Chayton's 36 deposits? F. 710 dollar G. 1,850 dollar H. 6,300 dollar J. 6,480 dollar K. 6,660 dollar

Short answer

Answer: K. 6,660 dollars

Step-by-step solution

Identifying an arithmetic series

Chayton's deposits are increasing by 10 dollars every month: 10, 20, 30, ..., 360. This progression is an arithmetic series.

Write the formula for the sum of an arithmetic series

To find the sum of an arithmetic series, we use the formula: Sum = n * (a_1 + a_n) / 2 where n is the number of terms, a_1 is the first term, and a_n is the last term.

Determine the number of terms and corresponding values

In Chayton's deposit series, the number of terms n is 36, the first term a_1 is 10, and the last term a_36 is 360.

Calculate the sum of deposits

Using the values from Step 3, we can find the sum: Sum = 36 * (10 + 360) / 2 Sum = 36 * 370 / 2 Sum = 18 * 370 Sum = 6660

Compare the sum with the answer choices

The total amount of Chayton's 36 deposits is 6,660 dollars. Comparing it with the answer choices, the correct answer is: K. 6,660 dollar

Key concepts

Mathematics

Mathematics is a vast field that involves numbers, calculations, and an array of numerical concepts, one of which is arithmetic series. Understanding arithmetic series is crucial for solving a variety of real-life problems, such as calculating the total savings over time, as seen in Chayton's example.
An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which each term after the first is obtained by adding a constant difference, known as the common difference. The concept of arithmetic series applies to diverse areas in mathematics such as algebra, financial calculations, and even higher-level math topics like calculus and beyond. Having a firm grasp of this concept lays the groundwork for more complex mathematical problem-solving.

ACT Prep

The ACT exam includes a variety of math concepts, and proficiency in arithmetic series can contribute significantly to a student's score. Exam questions may present scenarios similar to Chayton's savings, requiring the examinee to apply formulas and critical thinking to compute sums.
To prepare for the ACT and similar standardized tests, familiarizing oneself with the process of identifying an arithmetic series and using the sum formula is not only beneficial but often necessary. Practice with a wide range of problems can help solidify these concepts. Moreover, understanding the step-by-step method illustrated in Chayton's savings can boost confidence and improve time management during the exam.

Series and Sequences

When studying series and sequences in mathematics, it's essential to differentiate between the two. A sequence is a list of numbers in a specific order, while a series refers to the sum of the elements of a sequence. Arithmetic series, therefore, is the sum of a sequence where each term increases by a fixed amount.
The sum of an arithmetic series can be calculated as follows:

Sum = \( n \times \frac{{a_1 + a_n}}{2} \)

where \( n \) represents the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. This formula is derived from the principle that the average of the first and last term, multiplied by the number of terms, gives the total sum of the series. Understanding how this formula works is pivotal for correctly calculating the series total and uncovering patterns within sequences.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is created by adding a constant number, also known as the common difference, to the previous term. In Chayton's deposit scenario, the progression is the increasing amount of money he saves each month, and the common difference is 10 dollars.
These progressions are everywhere in daily life — from saving money, to the steps you climb, to the pattern of seats in a theater. Learning to identify and calculate the sum of an arithmetic progression not only aids in academic pursuits but also elevates one's ability to analyze and solve practical problems. Recognizing this pattern and utilizing the sum formula allows individuals to quickly determine the cumulative effect of repeated increments, important in financial planning, construction, scheduling, and other strategic activities.