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Express 0.overset(.)4 as an infinite geo...

Express `0.overset(.)4` as an infinite geometric series and hence prove that `(0.444…)^(1//2) = 0.666…`

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The correct Answer is:
`(4)/(10) + (4)/(10^(2)) + (4)/(10^(3)) +…oo`
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CHHAYA PUBLICATION-SEQUENCE AND SERIES-Exercise 9 D (Very Short Answer Type Questions)
  1. Find the sum of each of the following infinite geometric series, if it...

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  2. Express each of the following recurring decimals in the form of an inf...

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  3. Express each of the following recurring decimals in the form of an inf...

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  4. Express each of the following recurring decimals in the form of an inf...

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  5. Express each of the following recurring decimals in the form of an inf...

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  6. Express 0.overset(.)4 as an infinite geometric series and hence prove ...

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  7. The sum of an infinite geometric series is (1)/(3) and its first terms...

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  8. The sum of an infinite geometric series is 6 and the sum of its first ...

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  9. In a G.P. the sum of infinite terms is 15, the sum of the squares of t...

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  10. Two infinite geometric series starts from the same number. If the comm...

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  11. The distance passed over by a certain pendulum bob in succeeding swing...

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  12. In an infinite G.P., each term is equal to three times the sum of all ...

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  13. If x = a + (a)/(r ) + (a)/(r^(2))+…oo (-1 lt (1)/(r ) lt 1), y = b -...

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  14. If |a| lt 1, |b| lt 1, then show that a(a+b) + a^(2) (a^(2) + b^(2))...

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  15. If, for real numbers a, b, c, r with |r| gt 1, x = a + (a)/(r ) + (a)/...

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  16. If for two reals a, b, with |a| gt 1, |b| gt 1, x = 1 + (1)/(a) + (1)/...

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  17. For a real number 'a' with |a| lt 1, if x = a - a^(3) + a^(5)-…, y = 1...

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  18. For a real number a with 0 lt a lt (1)/(2), if b = 1 - a + a^(2) - a^(...

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  19. If, for 0 lt a lt (1)/(2), the sum of series a+a^(2) + a^(3)+… is b, t...

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  20. If x = underset(n = 0)overset(oo)Sigma cos^(2n)theta, " " y = underset...

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