logic and sequences sat 7.5
TRANSCRIPT
LOGIC AND SEQUENCES
“PRACTICE” FOR THE QUIZ7.5
LOGICproblem # 1
7+4+x+y+5 =32 and 7+4+x+z+5 =52 find the value for z-y.
SOLUTION #1
In the first problem, 16 + x + y = 32 or x + y = 16.
In the second, 16 + x + z = 52 or x+z = 36.
Solve for y: 16-x and solve for z: 36-x.
Then z-y = 36-x –(16-x) = 20
LOGICproblem # 2
Suppose 0.XY + 0.YX 0.XX X AND Y ARE DIGITS. What must Y
be?
SOLUTION #2
Think: I am adding Y to X but getting X back. It is the additive identity…which is ??? 0
SEQUENCES problem #3
32, 16, 8, 4 … What is the 8th term?
SOLUTION #3
Look for a pattern! Each number is the result of the preceding # being divided by 2, called a geometric sequence. Continue to get 2,1,1/2, ¼. The answer is ¼.
LOGICproblem #4
Suppose an “m-triple” is defined as (m/2 , m , 2m) for some number m.
Find an “m- triple” for the m value 60.
SOLUTION #4
Just substitute to find the triple. 60/2 and 2(60)
The triple would be: (30, 60, 120)
LOGICproblem #5
Let k # m be defined as the sum of all integers between k and m. For example: 6 # 9 = 7 + 8 = 15.
Find the value of ( 20 # 30) - ( 21 # 29)!
SOLUTION #5
Make a list: (21 + 22 +23 ….+29) – ( 22 +23 +…+28). Notice that all will “subtract out”
except for the 21 and the 29, so 21+29 = 50
SEQUENCESproblem #6
If a sequence has its nth term as 2n+3, find the sum of its third and sixth terms.
SOLUTION #6
The sequence is found by substituting in 1, then 2, then 3, etc to find the terms. So: 5,7,9,11,13,15. Now add terms 3 and 6: 9+ 15 =24
SEQUENCESproblem #7
Find the 10th term in the sequence: 1,2,3,5,8,13…
SOLUTION #7
Close inspection yields that a term is found by adding 2 previous terms: 1+2 =3; 2+3=5, etc so the 10th term is found after generating: 21, 34, 55, 89. The 10th term is 89.
LOGICproblem #8
If a is a nonnegative number and a* is defined as a* = sqrt(a) /3. When a* = 2, what must a be?
SOLUTION #8
Working backwards: sqrt(a) /3 = 2 says sqrt(a) would be 6. Therefore a = 36!
LOGICproblem #9
If r▲ t = rt – t for all integers r and t, find 6 ▲ 4.
SOLUTION #9
6 ▲ 4 = 6(4) -4 24 – 4 20
LOGICproblem #10
For x> 0, ♣ x is defined as 2x -1. What is the value of ♣6 / ♣3?
SOLUTION # 10
♣6 = 2(6) -1 =11 ♣3 = 2(3) -1 =5 11/5