From the odd number 19, 19 × 3 + 1 = 58. Dividing by 2 gives 29. Since 29 is odd, 29 × 3 + 1 = 88. Then successively dividing by 2 gives 44, 22, and 11.

For the odd number 11, 11 × 3 + 1 = 34. Half of 34 is 17. For the odd number 17, 17 × 3 + 1 = 52. Dividing by 2 twice produces 26 and then 13.

From 13, 13 × 3 + 1 = 40. Successive divisions by 2 give 20, 10, and 5. Then 5 × 3 + 1 = 16. Halving repeatedly gives 8, 4, 2, and finally 1. Thus, the sequence successfully reaches 1.

Q8. Write ... sum is $18{,}670$ Ans: $17{,}000 + 800 + 870 = 18670$ --- Q9. Check if ... $100$ Ans: $\frac{100}{2} = 50$ $\frac{50}{2} = 25$ $25 \times 3 + 1 = 76$ $\frac{76}{2} = 38$ $\frac{38}{2} = 19$ $19 \times 3 + 1 = 58$ $\frac{58}{2} = 29$ $29 \times 3 + 1 = 88$ $\frac{88}{2} = 44$ $\frac{44}{2} = 22$ $\frac{22}{2} = 11$ $11 \times 3 + 1 = 34$ $\frac{34}{2} = 17$ $17 \times 3 + 1 = 52$ $\frac{52}{2} = 26$ $\frac{26}{2} = 13$ $13 \times 3 + 1 = 40$ $\frac{40}{2} = 20$ $\frac{20}{2} = 10$ $\frac{10}{2} = 5$ $5 \times 3 + 1 = 16$ $\frac{16}{2} = 8$ $\frac{8}{2} = 4$ $\frac{4}{2} = 2$ $\frac{2}{2} = 1$

Answer: Yes, following the Collatz rule starting from $100$ , the sequence reaches $1$ in $25$ steps: $100 \to 50 \to 25 \to 76 \to 38 \to 19 \to 58 \to 29 \to 88 \to 44 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$ .

Step-by-step solution

Step 1: State the rule and start at 100

The Collatz process takes any positive integer $n$ . If $n$ is even, we divide it by $2$ ; if $n$ is odd, we multiply it by $3$ and add $1$ . We repeat this procedure starting with $n = 100$ .

Step 2: Compute terms from 100 to 19

Starting at $100$ , dividing by $2$ gives $50$ , and dividing again gives $25$ . Since $25$ is odd, we compute $25 \times 3 + 1 = 76$ . Then $76 \div 2 = 38$ , and $38 \div 2 = 19$ .

Step 3: Compute terms from 19 to 11

From the odd number $19$ , $19 \times 3 + 1 = 58$ . Dividing by $2$ gives $29$ . Since $29$ is odd, $29 \times 3 + 1 = 88$ . Then successively dividing by $2$ gives $44$ , $22$ , and $11$ .

Step 4: Compute terms from 11 to 13

For the odd number $11$ , $11 \times 3 + 1 = 34$ . Half of $34$ is $17$ . For the odd number $17$ , $17 \times 3 + 1 = 52$ . Dividing by $2$ twice produces $26$ and then $13$ .

Step 5: Compute terms from 13 to 1

From $13$ , $13 \times 3 + 1 = 40$ . Successive divisions by $2$ give $20$ , $10$ , and $5$ . Then $5 \times 3 + 1 = 16$ . Halving repeatedly gives $8$ , $4$ , $2$ , and finally $1$ . Thus, the sequence successfully reaches $1$ .

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