Using mathematical induction, prove or disprove that all checkerboards of these shapes can be completely covered using right triominoes whenever n is a positive integer.

a) 3 x 2^n
b) 6 x 2^n
c) 3^n x 3^n
d) 6^n x 6^n

Solution Preview

Two right triominoes will exactly cover an area of the checkerboard of dimensions 2x3. Therefore, if the total number of squares on the checkerboard is evenly divisible by 6, it will be possible to completely cover the checkerboard with right triominoes.

To use mathematical induction, we first test the case n = 1 and then test the case n + 1. If both cases are true, the statement is proven and the checkerboard can be covered.

a) Using 1:
Number of squares = ...

Solution Summary

This solution uses mathematical induction to prove or disprove that four checkerboards of different dimensions can be completely covered by right triominoes. All calculations are shown.

Use mathematical induction to show that a rectangular checkerboard with an even number of cells and two cells missing, one white and one black, can be covered by dominoes.

CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter's life. Mr. Brown

Get your checkerboard and place one grain of wheat on the first square. Then place two grains of wheat on the next square. Then place four grains on the third square. Continue this until all 64 squares are covered with grains of wheat." As he had just harvested his wheat, Mr. Crane did not consider this much of an award, but he

Please help me with the attached pre-calculus questions.
1. Use mathematical induction to prove that 6 is a factor of n (n + 1)(n + 2).
2. Use mathematical induction to prove 1^2 + 3^2 + 5^2 + . . . + (2n - 1)^2 = n (4n^2 - 1) / 3.
3. Expand (2x - y)^6 using binomial coefficients and then evaluate each coefficient.
Ex

Let y1=1,and for each n belong to N define y_n+1=(3y_n+4)/4.
a-Use induction to prove that the sequence satisfies y_n<4for all n
belong to N.
b-use another induction argument to show the sequence(y1,y2,y3,...)is
increasing.

Application of Mathematical Induction
Application of Mathematical Induction
Fibonacci Numbers :- The Fibonacci numbers are numbers that has the following properties.
If Fn represents the nth Fibonacci number,
F1 = 1, F2 =1, F3 =2, F4=3, F5 = 5 etc.
We can find the Fibonacci number