practice and problem solving
find the next item in each pattern.

8 a.m., 11 a.m., 2 p.m., ... 12. 75, 64, 53, ... 13. △, □, ○, ...

complete each conjecture.

a rule in terms of n for the sum of the first n even positive integers is _?_.

the number of nonoverlapping segments formed by n collinear points is _?_.

industrial arts about 5% of the students at lincoln high school usually participate in the robotics competition. there are 526 students in the school this year. make a conjecture about the number of students who will participate in the robotics competition this year.

show that each conjecture is false by finding a counterexample.

if 1 - y>0, then 0 < y < 1.

for any real number x, x³ ≥ x².

every pair of supplementary angles includes one obtuse angle.

make a conjecture about each pattern. write the next two items.

2, 4, 16, ... 21. 1/2, 1/4, 1/8, ... 22. -3, 6, -9, 12, ...

draw a square of dots. make a conjecture about the number of dots needed to increase the size of the square from n×n to (n + 1)×(n + 1).

2 - 1 using inductive reasoning to make conjectures 77

Question

practice and problem solving
find the next item in each pattern.
11. 8 a.m., 11 a.m., 2 p.m., ... 12. 75, 64, 53, ... 13. △, □, ○, ...
complete each conjecture.
14. a rule in terms of n for the sum of the first n even positive integers is _?_.
15. the number of nonoverlapping segments formed by n collinear points is _?_.
16. industrial arts about 5% of the students at lincoln high school usually participate in the robotics competition. there are 526 students in the school this year. make a conjecture about the number of students who will participate in the robotics competition this year.
show that each conjecture is false by finding a counterexample.
17. if 1 - y>0, then 0 < y < 1.
18. for any real number x, x³ ≥ x².
19. every pair of supplementary angles includes one obtuse angle.
make a conjecture about each pattern. write the next two items.
20. 2, 4, 16, ... 21. 1/2, 1/4, 1/8, ... 22. -3, 6, -9, 12, ...
23. draw a square of dots. make a conjecture about the number of dots needed to increase the size of the square from n×n to (n + 1)×(n + 1).
2 - 1 using inductive reasoning to make conjectures 77

Accepted Answer

### 11. # Explanation: ## Step1: Find time - interval The time - interval from 8 A.M. to 11 A.M. is 3 hours, and from 11 A.M. to 2 P.M. is also 3 hours. ## Step2: Calculate next time 2 P.M.+3 hours = 5 P.M. # Answer: 5 P.M. ### 12. # Explanation: ## Step1: Find the common difference 64 - 75=-11, 53 - 64=-11. The common difference is - 11. ## Step2: Calculate next number 53+( - 11)=42 # Answer: 42 ### 13. # Explanation: ## Step1: Observe the pattern The pattern seems to be rotating the shape 90 degrees clock - wise each time. ## Step2: Determine next shape The next shape after ⬚ (a circle - like shape) rotated 90 degrees clock - wise is a rotated triangle. # Answer: A rotated triangle ### 14. # Explanation: ## Step1: List the first few even positive integers The first even positive integer is 2, the sum of the first 1 even positive integer $S_1 = 2=1\times(1 + 1)$; the sum of the first 2 even positive integers $2+4 = 6=2\times(2 + 1)$; the sum of the first 3 even positive integers $2 + 4+6=12 = 3\times(3 + 1)$. ## Step2: Generalize the rule The sum of the first $n$ even positive integers $S_n=n(n + 1)$ # Answer: $n(n + 1)$ ### 15. # Explanation: ## Step1: Consider the case for a small number of collinear points For 2 collinear points, the number of non - overlapping segments is 1; for 3 collinear points, the number of non - overlapping segments is 2; for 4 collinear points, the number of non - overlapping segments is 3. ## Step2: Generalize the rule The number of non - overlapping segments formed by $n$ collinear points is $n-1$ # Answer: $n - 1$ ### 16. # Explanation: ## Step1: Calculate the number of participants We know that 5% of 526 students will participate. $5\%=\frac{5}{100}=0.05$, and $0.05\times526 = 26.3\approx26$ (since we are talking about the number of students, we round to the nearest whole number). # Answer: 26 ### 17. # Explanation: ## Step1: Solve the inequality $1 - y>0$ Subtract 1 from both sides: $-y>-1$. Multiply both sides by - 1 (and reverse the inequality sign), we get $y < 1$. A counter - example is $y=-1$, since $1-(-1)=2>0$, but $y=-1$ is not in the interval $(0,1)$. # Answer: $y=-1$ ### 18. # Explanation: ## Step1: Test a value of $x$ Let $x = 0.5$. Then $x^3=(0.5)^3=0.125$ and $x^2=(0.5)^2 = 0.25$, and $0.125<0.25$, so $x^3 ot\geq x^2$ when $x = 0.5$ # Answer: $x = 0.5$ ### 19. # Explanation: ## Step1: Recall the definition of supplementary angles Two angles are supplementary if their sum is $180^{\circ}$. A counter - example is two right angles. Since $90^{\circ}+90^{\circ}=180^{\circ}$, and a right angle is not an obtuse angle. # Answer: Two right angles ### 20. # Explanation: ## Step1: Find the pattern $2^2 = 4$, $4^2=16$. The pattern is that each number is the square of the previous number. ## Step2: Calculate next two numbers $16^2 = 256$, $256^2=65536$ # Answer: 256, 65536 ### 21. # Explanation: ## Step1: Identify the pattern The denominator is doubling each time. The general term of the sequence is $\frac{1}{2^n}$, where $n = 1,2,3,\cdots$. ## Step2: Calculate next two terms When $n = 4$, the term is $\frac{1}{16}$; when $n = 5$, the term is $\frac{1}{32}$ # Answer: $\frac{1}{16},\frac{1}{32}$ ### 22. # Explanation: ## Step1: Analyze the sign and magnitude The signs alternate, and the magnitudes are multiples of 3. The general term is $(-1)^n\times3n$ ## Step2: Calculate next two terms When $n = 5$, $(-1)^5\times3\times5=-15$; when $n = 6$, $(-1)^6\times3\times6 = 18$ # Answer: - 15, 18 ### 23. # Explanation: ## Step1: Calculate the number of dots in $n\times n$ and $(n + 1)\times(n + 1)$ squares The number of dots in an $n\times n$ square is $n^2$, and the number of dots in an $(n + 1)\times(n + 1)$ square is $(n + 1)^2=n^2+2n + 1$. ## Step2: Find the difference $(n + 1)^2-n^2=2n + 1$ # Answer: $2n + 1$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

11.

Step1: Find time - interval

Step2: Calculate next time

Step1: Find the common difference

Step2: Calculate next number

Step1: Observe the pattern

Step2: Determine next shape

Answer:

12.