1. Create your own game or task that includes any of the following concepts (you can choose one or more):
ARITHMETIC MEAN
ARITHMETIC SEQUENCE
ARITHMETIC SERIES
GEOMETRIC SEQUENCE
GEOMETRIC MEAN
GEOMETRIC SERIES
FIBONNACI SEQUENCE
HARMONIC SEQUENCE
2. Criteria;
The Correct Answer and Explanation is :
Game Name: “Treasure Hunt: Sequences and Means”
Objective:
Find the treasure by solving clues related to sequences and means. Each correct answer unlocks the next clue until the treasure is found.
Task:
You are a treasure hunter searching for a lost artifact. To find it, you must solve a series of puzzles, each based on a mathematical concept. Here’s the first puzzle:
Clue:
The coordinates of the treasure are hidden in an arithmetic sequence. The sequence starts at 5, and each term increases by 7.
Find the 10th term in the sequence.
Hint: Use the formula for the ( n )-th term of an arithmetic sequence:
[ a_n = a_1 + (n-1)d ]
Where ( a_1 ) is the first term, ( d ) is the common difference, and ( n ) is the term number.
Solution:
Using the formula:
[ a_{10} = 5 + (10-1)(7) ]
[ a_{10} = 5 + 63 ]
[ a_{10} = 68 ]
The 10th term is 68.
Explanation (300 words):
An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a fixed amount, called the “common difference,” to the previous term. In this problem, the sequence starts at 5 (( a_1 = 5 )), and the common difference is 7 (( d = 7 )).
To find the ( n )-th term of an arithmetic sequence, we use the formula:
[ a_n = a_1 + (n-1)d ]
Here, ( n = 10 ), so we plug in the values:
[ a_{10} = 5 + (10-1)(7) ]
[ a_{10} = 5 + 63 ]
[ a_{10} = 68 ]
This formula works because it systematically adds the common difference ( d ) for ( (n-1) ) times, accounting for all the terms between the first and the ( n )-th. Understanding arithmetic sequences is critical in fields like finance, physics, and computer science, where patterns and predictability matter.
In this treasure hunt game, solving the arithmetic sequence led us to a term that could represent a coordinate, a password, or even a step in the journey. This approach ties math to real-world problem-solving, making it engaging and practical.