The numbers 49,4489,444889,.... obtained by inserting 48 into the middle of the preceding numbers are square of integers. (a) true or (b) false. explain

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.