Prove that in a sequence of numbers 49,4489,444889,44448889 in which every number is made by inserting 48-48 in the middle of previous as indicated, each number is the square of an integer.

Explanation

Given numbers are 49, 4489, 444889,…
Then nth number is `t_(n)=underset("n times")ubrace(444............4)underset("(n-1)times")ubrace(888.........8)9`
`=9+(8xx10+8xx10^(2)+….8xx10^(n-1))+(4xx10^(n)+4xx10^(n+1)+…4xx10^(2n-1))`
`=9+80((10^(n-1)-1)/(10-1))+4xx10^(n)((10^(n)-1)/(10-1))`
`=(81+80xx10^(n-1)-80+4xx10^(2n)-4xx10^(n))/9`
`=(1+8xx10^(n)+4xx10^(2n)-4xx10^(n))/9`
`=(1+4xx10^(n)+4xx10^(2n))/9`
`=((2xx10^(2n)+1)/(3))^(2)`, which is a perfect square of an integer.