Understand the problem
Find the sum 1+111+11111+1111111+…..1….111(2k+1) ones
Source of the problem
C.M.I UG-2019 entrance exam
Topic
Algebra
Difficulty Level
3.5 out of 10
Suggested Book
challenges and trills of pre college mathematics
Start with hints
Do you really need a hint? Try it first!
can you some how manipulate it in geometric progression .
\(\frac{1}{9}(9+99+999+9999………)\)
now can u transfer it into G.P series
ok let’s see
\(\frac{1}{9}[(10^1-1)+(10^2-1)+………] \) upto 2k+1 terms
= \(\frac{1}{9}[(10+10^2+10^3+……)-(1+1+1+1…….)]\)
now can u see the G.P Series , with first term 10 and common ratio 10
use the formula ,\( \frac{a(1-r^n)}{r-1}\)
so the final ans is \(\frac{1}{9}[\frac{10}{9}(10^{2k+1}-1)-n]\)
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