The sum of the first 10 terms of the series $9+99+999+.....$ is
- $\cfrac { 9 }{ 8 } \left( { 9 }^{ 10 }-1 \right) $
- $\cfrac { 100 }{ 9 } \left( { 10 }^{ 9 }-1 \right) $
- ${ 10 }^{ 9 }-1$
- $\cfrac { 100 }{ 9 } \left( { 10 }^{ 10 }-1 \right) $
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Correct answer
AI explanation
We can rewrite each term of the series as a power of 10 minus 1, changing the sequence into (10 - 1) plus (100 - 1) plus (1000 - 1), continuing to 10 terms. This separates into the geometric series 10 plus 100 plus 1000, continuing to the 10th term, minus 10 ones. The geometric sum equals 10 multiplied by the quantity 10 to the power of 10 minus 1, divided by 9, and subtracting 10 gives 10 to the power of 10 minus 100, all divided by 9. Factoring out 100 from the numerator yields 100 divided by 9 multiplied by the quantity 10 to the power of 9 minus 1.