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Math Help - One more for the road (number series)

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

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  2. #2
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    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.)

    . . 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.

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




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

    . . 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.

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


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  3. #3
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    Got another one!


    (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.

    \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 \\<br />
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}
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  4. #4
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    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 2

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

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

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

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

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

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

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

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

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

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

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

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

    \cdots
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