Appearance
question:The Fibonacci sequence is the sequence 1, 1, 2, X, 5, ldots where the first and second terms are 1 and each term after that is the sum of the previous two terms. The remainder when the 100^{mathrm{th}} term of the sequence is divided by 8 is 3. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the value of x, which represents the fourth term of the Fibonacci sequence. We know that the first and second terms of the Fibonacci sequence are 1. Each term after that is the sum of the previous two terms. The remainder when the 100th term of the sequence is divided by 8 is 3. Let's find the remainder when the 100th term is divided by 8: To find the remainder when a number is divided by 8, we can look for a pattern in the remainders of the terms. The remainders of the terms in the Fibonacci sequence repeat every 6 terms. The remainders of the terms in the Fibonacci sequence when divided by 8 are: 1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0, 1, 1, 2, 3, ... We can see that the remainder of the 100th term is 3. Therefore, the fourth term of the Fibonacci sequence is 3. The value of x is 3. The answer is: 3
question:Coach Grunt is preparing the 5-person starting lineup for his basketball team, the Grunters. There are 12 players on the team. Two of them, Ace and Zeppo, are league All-Stars, so they'll definitely be in the starting lineup. How many different starting lineups are possible? (The order of the players in a basketball lineup doesn't matter.)
answer:There are 12 players on the team, and 2 of them are already chosen to be in the starting lineup. So we need to choose 3 more players from the remaining 10 players to complete the lineup. The number of ways to choose 3 players from 10 is binom{10}{3}=frac{10!}{3!7!}=120. Therefore, there are boxed{120} different starting lineups possible. The answer is: 120
question:f Jemma initially saw 7 grasshoppers on her African daisy plant and then found 2 dozen baby grasshoppers hopping on the grass under the plant, how many grasshoppers did she find in total?
answer:One dozen is equal to 12, so 2 dozen is equal to 2 * 12 = 24. Jemma initially saw 7 grasshoppers and found 24 baby grasshoppers, so in total she found 7 + 24 = 31 grasshoppers. 31 The answer is: 31
question:Zack has traveled to twice the number of countries Patrick traveled to. Patrick traveled to three times the number of countries Joseph traveled to. Joseph traveled to half the number of countries George traveled to. If George traveled to x countries, Zack traveled to 18 countries. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the number of countries George traveled to. Let's break down the information given: Number of countries George traveled to: x Number of countries Joseph traveled to: 1/2 * x Number of countries Patrick traveled to: 3 * (1/2 * x) Number of countries Zack traveled to: 2 * (3 * (1/2 * x)) We know that Zack traveled to 18 countries. Therefore: 2 * (3 * (1/2 * x)) = 18 Let's simplify and solve for x: 2 * (3 * (1/2 * x)) = 18 2 * (3 * x/2) = 18 2 * (3x/2) = 18 To simplify further, we can cancel out the 2 and 2/2: 3x = 18 To isolate x, we divide both sides of the equation by 3: 3x/3 = 18/3 x = 6 The value of x is 6. 6 The answer is: 6