Question

name:date:directions: study each pattern carefully. use inductive reasoning to predict the next terms or make a conjecture. show your reasoning whenever possible.bell work reviewpart a: extend the pattern. write the next term1) 7, 14, 28, 56, __2) 2, 4, 8, 16, 32, 3) 11, 22, 33, 44, __4) ★, ★★, ★★★, ★★★★, ______you used data to find patterns and write the next term.part c: real-life pattern5) a caterpillar grows 2 cm each day.day 1 = 2 cm, day 2 = 4 cm, day 3 = 6 cm.predict: what will its length be on day 5? ____1. ex 1. write a conjecture that describes the pattern 2, 4, 12, 48, 240. then use your conjecture to find the next item in the sequence.step1 look for a patternstep 2: make a conjectureanswer: __________2. ex 2. write a conjecture that describes the pattern shown. then use your conjecture to find the next item in the sequence.conjecture:3. ex 3. write a conjecture that describes the pattern in the sequence. then use your conjecture to find the next item in the sequence.$1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}$step1 look for a patternstep 2: make a conjecture. draw the next figure.answer:step1 look for a patternstep 2: make a conjectureanswer:

Explanation:

Part A: Extend the Pattern

Step1: Identify multiply-by-2 pattern

$7 \times 2 = 14$, $14 \times 2 = 28$, $28 \times 2 = 56$, so $56 \times 2 = 112$

Step2: Identify multiply-by-2 pattern

$2 \times 2 = 4$, $4 \times 2 = 8$, $8 \times 2 = 16$, $16 \times 2 = 32$, so $32 \times 2 = 64$

Step3: Identify add-11 pattern

$11 + 11 = 22$, $22 + 11 = 33$, $33 + 11 = 44$, so $44 + 11 = 55$

Step4: Identify +1 star per term pattern

1 star, 2 stars, 3 stars, 4 stars, so next is 5 stars: $\star\star\star\star\star$

Part C: Real-Life Pattern

Step5: Identify +2 cm per day pattern

Day 1: $2$, Day 2: $2+2=4$, Day 3: $4+2=6$, Day 4: $6+2=8$, Day 5: $8+2=10$

Ex 1: Sequence Conjecture & Next Term

Step6: Identify multiply-by increasing integer

$2 \times 2 = 4$, $4 \times 3 = 12$, $12 \times 4 = 48$, $48 \times 5 = 240$, so $240 \times 6 = 1440$
Conjecture: Each term is the previous term multiplied by the position of the term (starting at 2 for the second term).

Ex 2: Triangle Pattern Conjecture & Next Term

Step7: Identify incremental triangle count

Figure 1: 3 small triangles, Figure 2: $3 + 6 = 9$, Figure 3: $9 + 9 = 18$, so next add 12: $18 + 12 = 30$
Conjecture: The number of small triangles increases by a multiple of 3 (3, 6, 9, 12...) with each subsequent figure. The next figure is a large equilateral triangle divided into 30 small equilateral triangles (adding a row of 12 small triangles to the third figure).

Ex 3: Fraction Sequence Conjecture & Next Term

Step8: Identify reciprocal of squares pattern

$1 = \frac{1}{1^2}$, $\frac{1}{4} = \frac{1}{2^2}$, $\frac{1}{9} = \frac{1}{3^2}$, $\frac{1}{16} = \frac{1}{4^2}$, $\frac{1}{25} = \frac{1}{5^2}$, so next is $\frac{1}{6^2} = \frac{1}{36}$
Conjecture: Each term is the reciprocal of the square of its position in the sequence ($\frac{1}{n^2}$ where $n$ is the term number).

Answer:

Part A:

1. 112
2. 64
3. 55
4. $\star\star\star\star\star$

Part C:

1. 10 cm

Ex 1:
Conjecture: Each term is the previous term multiplied by its 1-based position (starting with ×2 for the second term).
Next term: 1440

Ex 2:
Conjecture: The number of small triangles increases by 3, 6, 9, 12,... (multiples of 3) with each figure.
Next figure: A large equilateral triangle subdivided into 30 small equilateral triangles (adding a 4th row of small triangles to the third figure), and the count is 30.

Ex 3:
Conjecture: Each term is $\frac{1}{n^2}$ where $n$ is the term's position.
Next term: $\frac{1}{36}$