Question

The sum of the first 10 terms if the series 9+99+999+.....is

Answer 1

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9 + 99 + 999 + ...upto 10 terms

= (10-1) + (100-1) + (1000-1) + ... 10 terms

= ( 10 + 100 + 1000 + ... 10 terms )
- ( 1 + 1 + 1 +... 10 terms )

= 10 + 10² + 10³ + ...10 terms - (10)

[ ∵ 1+1+1+...10 terms = 10×1 = 10 ]

= 10 [ (10^10 - 1) / (10-1) ] - 10

= 10 [ (10^10 - 1) / 9 ] - 10

= 10 [ (10^10 - 1 )/9 - 1 ]

= 10 [ (10^10 - 1 - 9) / 9 ]

= 10 [ (10^10 - 10) / 9 ]

= 10/9 × 10 (10^9 - 1)

= 100/9 (10^9 - 1)
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Note :

10 + 10² + 10³ + ...10 terms

★ It is a GP with first term, a = 10 common ratio, r = 10 and number of terms , n = 10

★ Formula Used :

Sum of n terms of GP = a(r^n - 1)/(r - 1)