Number Substitution( Week 1 Evaluation) 1. If a@b = |a+b|+|a-b|, what is the value of 7@(8@13)? 2. If a♣b = |a − 2b|, what is the value of (5♣3)♣8? 3. If a♠b = a2 − ab + b2, what is the value of (7♠5)♠6? 4. If a♥b = a + ab + b2, what is the value of (4♥3)♥6? 5. If a♦b = (a − 1) + ab + (b − 1)2, what is the value of (3♦2)♦4? 6. If 1⊜5 = 1+11+111+1111+11111, 2⊜4 = 2+22+222+2222, 3⊜3 = 3+33+333, and 4⊜2 = 4+44, what is the value of 7⊜2? 7. Observe that definitions: 1⊞2 = 3+4, 2⊞3 = 4+5+6, 3⊞4 = 5+6+7+8, 4⊞5 = 6+7+8+9+10, and 4⊞2 = 6+7. If x⊞3 = 36, what is x? 8. abc, acb, bac, bca, cab, cba
In the six triples of letters shown above, if a is in the middle of a triple, the triple is assessed as 1, if a is in the left of a triple, the triple is assessed as 0, and if a is in the right of a triple, the triple is assessed as 2. What is the sum of the values of the six triples? 9. If [X] means , what is the value of [[9] × [5]]? 10. Suppose Ω(1)=1 × 2 × 3, Ω(2)=2 × 3 × 4, Ω(3)=3 × 4 × 5, Ω(4)=4 × 5 × 6, ... and if − = . What is X? Express your result as a fraction in the lowest term. |