+45 Find the sum of all the natural numbers that are three - digit palindromes when expressed in base 5. Express your answer in base 5.
The smallest three-digit palindrome in base 5 is 1010, and the largest is 4444. The sum of all the natural numbers that are three-digit palindromes when expressed in base 5 is:
1010 + 1111 + 1212 + ... + 4444
We can factor out a 10 from each term:
10(10 + 11 + 12 + ... + 44)
The sum 10 + 11 + 12 + ... + 44 is a geometric series with first term 10 and common ratio 11/10. The sum of a geometric series is equal to the first term divided by 1 - the common ratio. In this case, the sum is equal to:
10 / (1 - 11/10) = 100
Therefore, the sum of all the natural numbers that are three-digit palindromes when expressed in base 5 is:
10(100) = 1000
In base 5, this is equal to 10000.
