Tyson was working with square numbers. He noticed a pattern for squaring consecutive
numbers that end in 5.
52 152 252 352
25 225 625 1225
a. Identify a pattern in the table.
b. Make a conjecture based on the pattern identified.
c. Test the conjecture.
d. Explain whether or not the conjecture will always work
a. The pattern in the table is that for any number ending in 5, its square always ends with 25.
b. Based on the pattern identified, a conjecture can be made that the square of any number ending in 5 will always end in 25.
c. Let's test the conjecture by squaring a few more numbers ending in 5:
- 75^2 = 5625
- 115^2 = 13225
- 185^2 = 34225
In each case, the squared value ends in 25, confirming the conjecture.
d. The conjecture will always work because multiplying any number ending in 5 by itself will result in a product that ends in 25 due to the multiplication of the last digit (5) with itself. This pattern holds true for all numbers ending in 5.
a. Pattern in the table: In the square numbers of consecutive numbers ending in 5, the units digit of the square is always 5, and the tens digit can be found by multiplying the tens digit of the original number by the next consecutive number.
b. Conjecture based on the pattern identified: When squaring consecutive numbers that end in 5, the resulting square number will always have a units digit of 5, and the tens digit will be the product of the tens digit of the original number and the next consecutive number.
c. Testing the conjecture: We can test the conjecture by squaring a few more consecutive numbers that end in 5 and checking if the pattern continues to hold.
For example, let's square 455 and 555:
- 455^2 = 207025, which follows the pattern: 2 is the tens digit (4 × 5), and 5 is the units digit.
- 555^2 = 308025, which also follows the pattern: 3 is the tens digit (5 × 6), and 5 is the units digit.
Both test cases support the conjecture.
d. Explanation of whether the conjecture will always work: The conjecture will always work for squaring consecutive numbers that end in 5. This is because when a number ends in 5, its square will always have a units digit of 5. The relationship between the tens digit of the original number and the next consecutive number remains consistent across all cases, leading to the pattern observed.
the differences are 200, 400, 600, ...
So work with that