Achieving Academic Success: Tackling a Challenging Quiz Problem
This post is designed for Google SEO to ensure it meets search engine optimization standards. The content is rich, informative, and designed to answer questions that students or teachers might have about solving quiz score problems. Let's dive into the details of the question and find the solution!
Introduction
Mathematics can sometimes be a challenging subject, but it can also be an excellent tool for understanding patterns and trends. One such problem involves the grade a student receives on a series of quizzes. In this case, a student began with a score of 55 on the first quiz and improved by 5 points on each subsequent quiz. Let's explore how to calculate the average of these seven quizzes and the steps involved.
The Problem
Here is the problem statement in detail:
A student's grade on the first of seven quizzes in Math was 55. However, on each successive quiz, her score was 5 more than on the preceding one. What was the average of the seven quizzes?
Understanding the Problem
This problem involves a sequence of numbers where each number is 5 points higher than the previous one. This type of sequence is known as an arithmetic progression (AP). In an AP, the difference between consecutive terms is constant.
Formulating the Solution
Let's break down the problem step by step:
Identify the first term (a) and the common difference (d): a 55 (the score on the first quiz) d 5 (each subsequent score is 5 higher than the previous one) N 7 (there are 7 quizzes in total)Using the Formula for the Sum of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
[text{Sum} frac{n}{2} times (2a (n-1)d)]Let's apply this formula to find the total score of all seven quizzes:
[text{Sum} frac{7}{2} times (2 times 55 (7-1) times 5)][text{Sum} frac{7}{2} times (110 30)]
[text{Sum} frac{7}{2} times 140]
[text{Sum} 490]
So, the total score for all seven quizzes is 490.
Calculating the Average
To find the average score, we need to divide the total score by the number of quizzes:
[text{Average} frac{text{Total Score}}{text{Number of Quizzes}}] [text{Average} frac{490}{7}] [text{Average} 70]Thus, the average score of the seven quizzes is 70.
Conclusion
By understanding the concept of arithmetic progressions, we can easily solve complex problems involving sequences of numbers. The key is to identify the first term, the common difference, and the number of terms. Using these values, we can apply the appropriate formulas to find the desired result. In this case, the average score of the seven quizzes is 70, which is a significant improvement from the initial score of 55.
This problem not only reinforces the importance of consistent practice and improvement but also demonstrates the practical application of mathematical concepts in real-world scenarios. Such exercises are essential for students and teachers alike, as they prepare for more complex problems and better understand the underlying principles that govern arithmetic progressions.