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# 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

Algebra

3.5 out of 10

##### Suggested Book

challenges and trills of pre college mathematics

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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