Riya's Enchanted Staircase A Math Adventure

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Introduction

In this enchanting tale, we follow Riya as she discovers a magical staircase with steps of varying heights. The step heights follow a peculiar pattern: 1, 3, 5, 7… This sequence immediately sparks our curiosity, hinting at a mathematical relationship lurking beneath the surface. Let's embark on this mathematical journey with Riya, unraveling the mysteries of this enchanted staircase. We'll explore the heights of the steps, the total height climbed, and even venture into predicting the characteristics of the 50th step. This exploration will not only reveal the beauty of mathematical sequences but also enhance our problem-solving skills. So, join us as we delve into the world of arithmetic progressions and discover the magic hidden within these seemingly simple steps. Understanding patterns is fundamental to mathematics, and this staircase provides an excellent real-world example to illustrate this concept. The problems we'll tackle will involve identifying patterns, applying formulas, and making logical deductions – all crucial skills in mathematical thinking. The enchanted staircase serves as a perfect backdrop for learning and applying mathematical concepts in an engaging and imaginative way. As we progress, we'll uncover not just the answers to the specific questions posed, but also a deeper appreciation for the elegance and power of mathematics in describing the world around us. So, let's begin our ascent and see where the mathematical staircase leads us!

1. What is the height of the 6th step?

To determine the height of the 6th step, we first need to recognize the pattern in the given step heights: 1, 3, 5, 7… This sequence represents an arithmetic progression, where the difference between consecutive terms is constant. In this case, the common difference is 2 (3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2). An arithmetic progression can be defined by the formula: a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference. Applying this formula to find the height of the 6th step (a_6), we have: a_1 = 1 (the height of the first step), n = 6 (we want to find the height of the 6th step), and d = 2 (the common difference). Substituting these values into the formula, we get: a_6 = 1 + (6 - 1) * 2 = 1 + 5 * 2 = 1 + 10 = 11. Therefore, the height of the 6th step is 11 units. This demonstrates the power of arithmetic progressions in predicting future terms in a sequence. The formula not only provides the answer but also a structured approach to solving similar problems. By identifying the pattern and applying the appropriate formula, we can efficiently calculate the height of any step in this enchanted staircase. Understanding arithmetic progressions is a key concept in algebra, with applications extending beyond simple sequences to more complex mathematical models. So, the 6th step stands tall at a height of 11, ready for Riya to climb!

2. What is the total height after the 7th step?

To find the total height climbed after the 7th step, we need to sum the heights of all the steps up to and including the 7th step. We already know the heights of the first four steps: 1, 3, 5, and 7. We also calculated the height of the 6th step to be 11. Following the arithmetic progression, the height of the 5th step would be 9, and the height of the 7th step can be calculated using the formula a_n = a_1 + (n - 1)d. For the 7th step (a_7), a_1 = 1, n = 7, and d = 2. So, a_7 = 1 + (7 - 1) * 2 = 1 + 6 * 2 = 1 + 12 = 13. Now we have the heights of all 7 steps: 1, 3, 5, 7, 9, 11, and 13. To find the total height, we sum these values: 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49. Alternatively, we can use the formula for the sum of an arithmetic series: S_n = (n/2)(a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term. In this case, n = 7, a_1 = 1, and a_7 = 13. Substituting these values, we get: S_7 = (7/2)(1 + 13) = (7/2)(14) = 7 * 7 = 49. Therefore, the total height climbed after the 7th step is 49 units. This illustrates two different approaches to solving the same problem, highlighting the versatility of mathematical tools. Whether we sum the individual heights or apply the arithmetic series formula, we arrive at the same answer. The enchanted staircase continues to reveal the beauty of mathematical relationships and the efficiency of mathematical formulas in solving real-world problems.

3. Is the 50th step odd or even?

To determine whether the 50th step is odd or even, we need to revisit the pattern of step heights. We know the sequence is 1, 3, 5, 7…, which consists entirely of odd numbers. Each term in this arithmetic progression is generated by adding 2 to the previous term, starting with 1. Since we are always adding an even number (2) to an odd number (the previous step height), the result will always be an odd number. This is because the sum of an odd number and an even number is always odd. We can also use the formula for the nth term of an arithmetic progression: a_n = a_1 + (n - 1)d. For the 50th step (a_50), a_1 = 1, n = 50, and d = 2. Substituting these values, we get: a_50 = 1 + (50 - 1) * 2 = 1 + 49 * 2 = 1 + 98 = 99. The result, 99, is an odd number. Therefore, the 50th step is odd. This demonstrates a fundamental property of arithmetic progressions with an odd first term and an even common difference – all terms will be odd. The enchanted staircase continues to reinforce the predictability and consistency of mathematical patterns. By understanding the underlying principles of arithmetic progressions, we can confidently determine the parity (whether it's odd or even) of any step in the sequence without having to calculate all the preceding steps. This highlights the power of mathematical reasoning in making generalizations and predictions.

4. What is the total height climbed after 50 steps?

To calculate the total height climbed after 50 steps, we need to find the sum of the arithmetic series consisting of the heights of the first 50 steps. We know the sequence of step heights is 1, 3, 5, 7…, which is an arithmetic progression with a_1 = 1 (the first term) and d = 2 (the common difference). We can use the formula for the sum of an arithmetic series: S_n = (n/2)(a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term. In this case, n = 50 (we want to find the sum of the first 50 terms), a_1 = 1, and we need to find a_50 (the height of the 50th step). We already calculated a_50 in the previous question: a_50 = 99. Now we can substitute these values into the formula for the sum: S_50 = (50/2)(1 + 99) = 25 * 100 = 2500. Therefore, the total height climbed after 50 steps is 2500 units. This calculation showcases the efficiency of the arithmetic series formula in finding the sum of a large number of terms. Instead of adding the heights of all 50 steps individually, we can use the formula to arrive at the answer quickly and accurately. The enchanted staircase, with its consistent arithmetic progression, provides a perfect example of how mathematical formulas can simplify complex calculations. This result highlights the significant height Riya would climb after 50 steps, emphasizing the cumulative effect of the step heights. The formula not only provides the answer but also a clear method for solving similar problems involving the sum of arithmetic series.

Conclusion

Riya's adventure with the enchanted staircase has been a fascinating journey into the world of mathematics. We've explored arithmetic progressions, calculated step heights, determined the parity of steps, and found the total height climbed. Through this exploration, we've not only answered specific questions but also gained a deeper understanding of mathematical patterns and formulas. The enchanted staircase served as a captivating context for learning about arithmetic progressions, demonstrating their relevance in real-world scenarios. We've seen how a simple sequence of numbers can lead to interesting and challenging problems, highlighting the power of mathematics in problem-solving. From calculating the height of the 6th step to finding the total height climbed after 50 steps, each question has reinforced the importance of identifying patterns, applying formulas, and making logical deductions. The journey with Riya has underscored the beauty and elegance of mathematics, showcasing its ability to describe and predict phenomena in the world around us. As we conclude this mathematical adventure, we carry with us not just the answers to the questions, but also a renewed appreciation for the power and versatility of mathematics. The enchanted staircase will remain a memorable example of how mathematical concepts can be explored and understood in an engaging and imaginative way. So, let's continue to explore the world with a mathematical lens, uncovering the patterns and relationships that shape our reality.