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BITSAT2013MathematicsSequences and SeriesActual

The sum of the series 1+2.2+3.2²+4.2³+ .+100.2⁹⁹ is

Options

  1. A100 2¹⁰⁰+1
  2. B99.2¹⁰⁰+1
  3. C99.2¹⁰⁰-1
  4. D100.2¹⁰⁰+1

Correct answer

B. 99.2¹⁰⁰+1

Step-by-step solution

Let S=1+2.2+3.2²+4.2²+ .+100.2⁹⁹ It is an arithmetico - geometric series. On multiplying Eq. (i) by 2 and then subtracting it from Eq. (i), we get S=1+2.2+3.2²+4.2³+ +100.2⁹⁹2 S=1.2+2.2² 3.2³+ +99.2⁹⁹+100.2¹⁰⁰ aligned -S &=1+2+2²+2³ +2⁹⁹-100.2¹⁰⁰ & -S= 1 (2¹⁰⁰-1 ) 2-1 -100.2¹⁰⁰ & -S=2¹⁰⁰-1-100.2¹⁰⁰ & -S=-1-99.2¹⁰⁰ & S=99.2¹⁰⁰+1 aligned

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