Questions: Complete each geometric sequence with the missing terms. Then find the growth factor for each. a. 5, 25 625 b. -1, -36, 216 c. 10, 5 0.625 d. 36, -108 e. 12, 18, 27 The first term of a sequence is 4. a. Choose a growth factor and list the next 3 terms of a geometric sequence, b. Choose a different growth factor and list the next 3 terms of a geometric sequence. Here is a rule that can be used to build a sequence of numbers once a starting number is chosen: Each number is two times three less than the previous number. a. Starting with the number 0, build a sequence of 5 numbers. b. Starting with the number 3, build a sequence of 5 numbers. c. Can you choose a starting point so that the first 5 numbers in your sequence are all positive? Explain your reasoning.

Solution Steps

For each geometric sequence, identify the common ratio (growth factor) by dividing a term by its preceding term. Use this ratio to fill in the missing terms.

a. Choose a growth factor and multiply the first term by this factor to get the next term. Repeat this process to get the next three terms. b. Choose a different growth factor and repeat the process to get the next three terms.

a. Start with the given number and apply the rule to generate the next number. Repeat this process to build the sequence. b. Start with the given number and apply the rule to generate the next number. Repeat this process to build the sequence.

For each geometric sequence, we identify the common ratio \( r \) and use it to fill in the missing terms.

• Common ratio: \( r = \frac{625}{5} = 125 \)
• Missing term: \( 5 \times 125 = 625 \)

Thus, the completed sequence is \( 5, 625, 625 \) with \( r = 125 \).

• Common ratio: \( r = \frac{-36}{-1} = 36 \)
• Missing term: \( -1 \times 36 = -36 \)

Thus, the completed sequence is \( -1, -36, -36, 216 \) with \( r = 36 \).

• Common ratio: \( r = \frac{5}{10} = 0.5 \)
• Missing term: \( 5 \times 0.5 = 2.5 \)

Thus, the completed sequence is \( 10, 5, 2.5, 0.625 \) with \( r = 0.5 \).

Given the first term \( a_1 = 4 \), we generate the next three terms for two different growth factors.

• Sequence: \( 4, 4 \times 2 = 8, 8 \times 2 = 16, 16 \times 2 = 32 \)

Thus, the sequence is \( 4, 8, 16, 32 \).

• Sequence: \( 4, 4 \times 0.5 = 2, 2 \times 0.5 = 1, 1 \times 0.5 = 0.5 \)

Thus, the sequence is \( 4, 2, 1, 0.5 \).

Given the rule: Each number is two times three less than the previous number, we generate sequences starting from different initial values.

• Sequence: \( 0, 2 \times 0 - 3 = -3, 2 \times -3 - 3 = -9, 2 \times -9 - 3 = -21, 2 \times -21 - 3 = -45 \)

Thus, the sequence is \( 0, -3, -9, -21, -45 \).

• Sequence: \( 3, 2 \times 3 - 3 = 3, 2 \times 3 - 3 = 3, 2 \times 3 - 3 = 3, 2 \times 3 - 3 = 3 \)

Thus, the sequence is \( 3, 3, 3, 3, 3 \).

Final Answer