1. ## 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 ???

2. 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.

3. This is an arithmetic sequence with first term 2 and common difference 5. Thus the nth term is

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

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.

4. $\displaystyle U_n = 2+5(n-1) = 5n-3 \implies n = \dfrac{U_n +3}{5}$

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

Trying $\displaystyle U_n = 20002$ gives $\displaystyle n = 4001$

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

Hence it would be 20,002

5. 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: .$\displaystyle abcd2\text{ or } pqrs7.$

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

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

It is obvious that the smallest number is: $\displaystyle 20002.$

6. Originally Posted by Soroban
Hello, bigwave!

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

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

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

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

It is obvious that the smallest number is: $\displaystyle 20002.$

Great solution!!!

7. thanks everyone that was a great help.

8. Originally Posted by Soroban
Hello, bigwave!

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

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

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

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

It is obvious that the smallest number is: $\displaystyle 20002.$

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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# the smallest 5 digit number which is palindrome using three digits

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