Here are the solutions to the sequences: Question 1: Write the next three numbers in each of the following sequences. Say whether each sequence is a constant difference or constant ratio sequence or neither. a) 3, 6, 9, 12, 15, ... Step 1: Find the difference between consecutive terms. $6-3=3$ $9-6=3$ $12-9=3$ $15-12=3$ The difference is constant (3). This is a constant difference sequence. Step 2: Add the constant difference to find the next three terms. $15+3=18$ $18+3=21$ $21+3=24$ The next three numbers are 18, 21, 24. The sequence is a constant difference sequence. $$\boxed{\text{18, 21, 24; Constant difference}}$$ b) 2, 4, 8, 16, ... Step 1: Find the ratio between consecutive terms. $\frac{4}{2}=2$ $\frac{8}{4}=2$ $\frac{16}{8}=2$ The ratio is constant (2). This is a constant ratio sequence. Step 2: Multiply by the constant ratio to find the next three terms. $16 \times 2=32$ $32 \times 2=64$ $64 \times 2=128$ The next three numbers are 32, 64, 128. The sequence is a constant ratio sequence. $$\boxed{\text{32, 64, 128; Constant ratio}}$$ c) 7, 11, 15, 19, 23, ... Step 1: Find the difference between consecutive terms. $11-7=4$ $15-11=4$ $19-15=4$ $23-19=4$ The difference is constant (4). This is a constant difference sequence. Step 2: Add the constant difference to find the next three terms. $23+4=27$ $27+4=31$ $31+4=35$ The next three numbers are 27, 31, 35. The sequence is a constant difference sequence. $$\boxed{\text{27, 31, 35; Constant difference}}$$ d) 1, 4, 9, 16, ... Step 1: Check for constant difference. $4-1=3$ $9-4=5$ $16-9=7$ The differences are not constant. Step 2: Check for constant ratio. $\frac{4}{1}=4$ $\frac{9}{4}=2.25$ The ratios are not constant. Step 3: Observe the pattern. The terms are squares of consecutive integers: $1^2, 2^2, 3^2, 4^2$. Step 4: Find the next three terms. $5^2=25$ $6^2=36$ $7^2=49$ The next three numbers are 25, 36, 49. The sequence is neither a constant difference nor a constant ratio sequence. $$\boxed{\text{25, 36, 49; Neither}}$$ Question 2: Describe the pattern in each of the following sequences. State whether it is a constant difference or constant ratio sequence or neither. Then fill in the missing numbers. a) 18, 36, 54, 72, [ ], [ ], [ ] Step 1: Find the difference between consecutive terms. $36-18=18$ $54-36=18$ $72-54=18$ The difference is constant (18). This is a constant difference sequence. Step 2: Add the constant difference to find the missing terms. $72+18=90$ $90+18=108$ $108+18=126$ The pattern is adding 18 to the previous term. The sequence is a constant difference sequence. The missing numbers are 90, 108, 126. $$\boxed{\text{Constant difference; 90, 108, 126}}$$ b) 3, 9, 27, [ ], [ ], [ ] Step 1: Find the ratio between consecutive terms. $\frac{9}{3}=3$ $\frac{27}{9}=3$ The ratio is constant (3). This is a constant ratio sequence. Step 2: Multiply by the constant ratio to find the missing terms. $27 \times 3=81$ $81 \times 3=243$ $243 \times 3=729$ The pattern is multiplying the previous term by 3. The sequence is a constant ratio sequence. The missing numbers are 81, 243, 729. $$\boxed{\text{Constant ratio; 81, 243, 729}}$$ *c) 1.25, 2.5, 3.75, 5.0, [ ], [ ], [