Lesson Plans SET B for Grades 9-10-Lesson 4 Sequences and Patterns HW

Lesson 4 Sequences and Patterns HW

Pdf Summary

This document provides practice problems and solutions from a Math Kangaroo class lesson on sequences and patterns, specifically aimed at levels 9-10. Below is a summary of the problems and their solutions:<br /><br />1. **Sequences of Multiples**: Find the 100th term in a sequence derived from numbers 1 to 1,000, keeping only those that are multiples of 3 or 7. The 100th term in this sequence is 234.<br /><br />2. **Circle Number Manipulation**: Starting with numbers from 1 to 1,776 in a circle, determine if a sequence of operations can reduce the numbers to 1. Since the total sum of initial numbers is even, and only even changes occur, the final number cannot be 1.<br /><br />3. **16-gon Vertex Numbers**: Numbers at vertices of a 16-gon are each the average of their two neighboring numbers. If one vertex, A, is 16, find the number at the opposite vertex B. Since all numbers end up being the same, the number at B is also 16.<br /><br />4. **Geometric Sequence**: Given the first four terms 1, 2, 3 of a geometric sequence, find the 50th term. Identifying the pattern as a geometric progression with a ratio, the 50th term is determined to be \(2^{49}\).<br /><br />5. **Circle Elimination Process**: <br />   - **(a)** Show for \( n = 2^m \), the last number left is n. Progressive elimination of odd numbers shows this pattern.<br />   - **(b)** For \( n=1999 \), find the last number. This example uses a specific competition problem.<br /><br />6. **Modified Circle Elimination**: Different starting point (starting from deleting the second number). For sequences from 1 to \(2n\), the final remaining number aligns with specific arithmetic rules. The method confirms results like when \( k = 32 \), it returns 1, demonstrating how to calculate the last number remaining for powers of 2 via a systematic elimination process.<br /><br />These problems and their corresponding solutions test understanding and application of concepts in sequences and pattern recognition, typical in competitive mathematics settings.

Keywords

Math Kangaroo

sequences

patterns

multiples

geometric sequence

circle elimination

vertex numbers

competitive mathematics

problem-solving

arithmetic rules