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Math Help - smallest five-digit palindrome

  1. #1
    Super Member bigwave's Avatar
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    smallest five-digit palindrome

    What term is the smallest five-digit palindrome in the arithmetic sequence 2,7,12,17...

    In that palindrome would be read the same way in either direction don't see how you get this from this sequence.

    would it be 97,102,97 ???
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  2. #2
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    That number is neither 5-digits nor is it a palindrome.
    10101 is a 5-digit palindrome. But I doubt it is in that AP.
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    This is an arithmetic sequence with first term 2 and common difference 5. Thus the nth term is

    a_n=2+5(n-1)=5n-3.

    The 5 digit palindromes in increasing order are 10001, 10101, 10201,...11011, 11111,...etc.

    I add 3 to each one in turn until I get a number divisible by 5. The first one that works is 20002.

    Remark: A systematic way to list palindromes of a fixed length in increasing order is to increase digits one by one from the middle outward as I have done above.
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    U_n = 2+5(n-1) = 5n-3 \implies n = \dfrac{U_n +3}{5}

    It therefore follows that, for n to be an integer, that the last digit of U_n is 2.

    Trying U_n = 20002 gives n = 4001

    Now testing it gives 2+5(4001-1) = 20002

    Hence it would be 20,002
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  5. #5
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    Hello, bigwave!

    What term is the smallest five-digit palindrome in the arithmetic sequence 2,7,12,17... ?

    A number of the sequence end in either 2 or 7.

    . . It has the form: . abcd2\text{ or } pqrs7.


    A palindrome in the sequence begins and ends with either 2 or 7.

    . . It has the form: . 2aba2 \text{ or } 7pqp7


    It is obvious that the smallest number is: 20002.

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  6. #6
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, bigwave!


    A number of the sequence end in either 2 or 7.

    . . It has the form: . abcd2\text{ or } pqrs7.


    A palindrome in the sequence begins and ends with either 2 or 7.

    . . It has the form: . 2aba2 \text{ or } 7pqp7


    It is obvious that the smallest number is: 20002.


    Great solution!!!
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  7. #7
    Super Member bigwave's Avatar
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    thanks everyone that was a great help.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by Soroban View Post
    Hello, bigwave!


    A number of the sequence end in either 2 or 7.

    . . It has the form: . abcd2\text{ or } pqrs7.


    A palindrome in the sequence begins and ends with either 2 or 7.

    . . It has the form: . 2aba2 \text{ or } 7pqp7


    It is obvious that the smallest number is: 20002.

    You need to point out that very positive number that ends in a 2 or a 7 is in the sequence.

    CB
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