# Thread: One more for the road (number series)

1. ## One more for the road (number series)

2. Hello, wonderboy1953!

Well, I got two of them, anyway . . .

[1] What are the next two numbers in this series?
. . .(Each number determines the next number.)

. . $\displaystyle 1, 4, 8, 13, 21, 30, 36, 45, 54, 63, 73, \_\_\,,\:\_\_$
Spoiler:

Each number is the preceding number plus the number of letters
. . in the preceding number.

$\displaystyle \begin{array}{ccccc} 1 + \text{"one"} &=& 1+3 &=& 4 \\ 4 + \text{"four"} &=& 4 + 4 &=& 8 \\ 8 + \text{"eight"} &=& 8 + 5 &=& 13 \\ 13 + \text{"thirteen"} &=& 13 + 8 &=&21 \\ \vdots && \vdots && \vdots \\ 63 + \text{"sixty-three"} &=& 63 + 10 &=& 73 \\ 73 + \text{"seventy-three"} &=& 73 + 12 &=& \boxed{85} \\ 85 + \text{"eighty-five"} &=& 85 + 10 &=& \boxed{95} \end{array}$

[7] What are the next two numbers in this series?
. . .The series in not mathematical. Some patriotism is required.)

. . $\displaystyle 1, 6, 10, 2,3,4,2,3,6,6,2,\:\_\_\,,\;\_\_$
Spoiler:

These are the number of letters in the words to the Pledge of Allegience.

. . $\displaystyle \begin{array}{cc} \text{I} & 1 \\ \text{pledge} & 6 \\ \text{allegience} & 10 \\ \text{to} & 2 \\ \text{the} & 3 \\ \text{flag} & 4 \\ \text{of} & 2 \\ \text{the} & 3 \\ \text{United} & 6 \\ \text{States} & 6 \\ \text{of} & 2 \\ \text{America} & \boxed{7} \\ \text{and} & \boxed{3} \end{array}$

3. Got another one!

$\displaystyle (2)\;\;30, 33, 42, 50, 55, 80, 88, 152, 162, 174,\:\_\_\,,\;\_\_$
Spoiler:

Each number is the preceding number
. . plus the product of the nonzero digits of the preceding number.

$\displaystyle \begin{array}{ccc} 30 \\ 30 + 3 &=& 33 \\ 33 + 3\!\cdot\!3 &=& 42 \\ 42 + 4\!\cdot\!2 &=& 50 \\ 50 + 5 &=& 55 \\ 55 + 5\!\cdot\!5 &=& 80 \\ 80 + 8 &=& 88 \\ 88 + 8\cdot8 &=& 152 \\ 152 + 1\!\cdot\!5\!\cdot\!2 &=& 162 \\ 162 + 1\!\cdot\!6\!\cdot\!2 &=& 174 \\ 174 + 1\!\cdot\!7\!\cdot\!4 &=& \boxed{202} \\ 202 + 2\!\cdot\!2 &=& \boxed{206} \end{array}$

4. Number Series 6

What are the next six numbers in this series? (Each number is determined from the number before by a simple mathematical relationship. You will need some math.)

2, 3, 4, 6, 8, 10, 13, 16, 20, 24, ?, ?, ?, ...
Spoiler:

Each number is the previous number plus the floor of the square root of that previous number.

$\displaystyle \displaystyle 2$

$\displaystyle \displaystyle 2 + \left\lfloor\sqrt{2}\right\rfloor = 3$

$\displaystyle \displaystyle 3 + \left\lfloor\sqrt{3}\right\rfloor = 4$

$\displaystyle \displaystyle 4 + \left\lfloor\sqrt{4}\right\rfloor = 6$

$\displaystyle \displaystyle 6 + \left\lfloor\sqrt{6}\right\rfloor = 8$

$\displaystyle \displaystyle 8 + \left\lfloor\sqrt{8}\right\rfloor = 10$

$\displaystyle \displaystyle 10 + \left\lfloor\sqrt{10}\right\rfloor = 13$

$\displaystyle \displaystyle 13 + \left\lfloor\sqrt{13}\right\rfloor = 16$

$\displaystyle \displaystyle 16 + \left\lfloor\sqrt{16}\right\rfloor = 20$

$\displaystyle \displaystyle 20 + \left\lfloor\sqrt{20}\right\rfloor = 24$

$\displaystyle \displaystyle 24 + \left\lfloor\sqrt{24}\right\rfloor = \boxed{28}$

$\displaystyle \displaystyle 28 + \left\lfloor\sqrt{28}\right\rfloor = \boxed{33}$

$\displaystyle \displaystyle 33 + \left\lfloor\sqrt{33}\right\rfloor = \boxed{38}$

$\displaystyle \cdots$