The 1025th term in this sequence is 1,22,4444,88888888,.......

Hello bonu !

Its a bit tricky ..!

See .. first term has ..1 digit ..1... 2^0

Second term has ....2 digit...22.....2^1

Third term has .....4 digits ....2^2

fourth term has ....8 digits ...2^3

So actually number of digits are in gp .. Hence in 1025 term ..we definitely have  2^1024  (isn't it ..! have doubt then ask)..
..and the digits of the number are actually the number of digit..interesting !
like ...22 ..(have two digit ..and digits are also 2 and 2)
4444  have 4 digits..and each digits are 4..
88888888  have 8 digits ..and each digit is 8 ..
The 1025 th term would have ..2^1024 digits ..and surely each digit would be ..2^1024 ..(like i explained)
So suppose  a=2^1024 ..
So 1025th term would be ..   (aaaaaa..........1024 times)..

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Lets leave this queation at this point .. and let me explain about this ..
any number can be written in GP format.. like ..
345 = 3X100 + 4X10 + 5X1   ( if instead of 3 ,4,5 ,,all digits are 3 ..then it become a perfect gp isn't it ? )
333= 3[100+10+1] ..
So we apply this concept here...
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We have ... aaaaaaa......1024 times ...

So it would be expressed as  .. = a [ 10^1023 + 10^1022 + 10^1021................+1^0].
aaaaaaaaaaaa...1024 times  =  a [ 1(10^1024-1)/(10-1)]    ( Sum of GP)
So the 1025 term would be .    (2^1024)(10^1024-1)/9   (Ans...)

Regards
Yagya