Each number is two times three less than the previous number

Question

1. Starting with the number 0, build a sequence of 5 numbers.
2. Starting with the number 3, build a sequence of 5 numbers.
3. Can you choose a starting point so that the first 5 numbers in your sequence are all positive? Explain your reasoning.

Answer 1

TWO times three less than the previous

2(x-3) where x is the previous

1) 0, -6, -18, -42, -90

2) 3, 0, -6, -18, -42

3) starts with x=10 --> 10, 14, 22, 38, 70

starts with x=6 ---> 6,6,6,6,6

Answer 2

X3 = 2(X2-3)

0=2(X2-3) =2X2-6

2X2 = 6

X2 = 3

3=2(X3-3) = 2X3-6

2X3 = 9

X3 = 9/2 = 4 1/2

9/2 = 2(X4-3) = 2X4 -6

2X4 = 21/2

X4 = 21/4 = 5 1/4

21/4 = 2(X4-3) = 2X4-6

21/4+24/4 = 2X

X = 45/8 = 5 5/8

0, 3, 4 1/2, 5 1/4, 5 4/8

3 =2(X2-3) = 2X2 -6

2X2 = 9

X2 = 9/2 = 4 1/2

9/2 = 2(X3-3) = 2X3 - 6

2X3 = 21/2

X3 = 21/4 = 5 1/4

.... repeats the first calculations starting at 0 until the 5th term

5 5/8 = 45/8 = 2(X5-3) = 2X5 -6

45/8 + 48/8 = 2X5

93/8 = 2X5

X5 = 93/16 = 5 13/16

sequence is

3, 4 1/2, 5 1/4, 5 5/8, 5 13/16

starting at 3 makes all numbers in the sequence positive