The value of the infinite series12 + 223! + 1

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 Multiple Choice QuestionsMultiple Choice Questions

61.

The value of

100011 × 2 + 12 ×3 + 13 ×4 + ... + 1999 × 1000

  • 1000

  • 999

  • 1001

  • 1999


62.

Six positive numbers are in GP, such that their product is 1000. If the fourth term is 1, then the last term is

  • 1000

  • 100

  • 1100

  • 11000


63.

Five numbers are in AP with common difference  0. If the 1st, 3rd and 4th terms are in GP, then

  • the 5th term is always 0.

  • the 1st term is always 0.

  • the middle term is always 0

  • the middle term is always - 2.


64.

The sum of series

11 × 2C025 + 12 × 3C125 + 13 × 4C225 + ... + 126 × 27C2525

is

  • 227 - 126 × 27

  • 227 - 2826 × 27

  • 12226 + 126 × 27

  • 226 - 152


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65.

Let f : R  R  be such that f is injective and f(x) f(y) = f(x + y) for all x, y if f(x), f(y) and f(z) are in GP, then x, y and z are in

  • AP always

  • GP always

  • AP depending on the values of x, y and z

  • GP depending on the values of x, y and z


66.

If P = 1 + 12 × 2 + 13 × 22 + ... and Q = 11 × 2 + 13 × 4 + 15 × 6 + ...,

then

  • P = Q

  • 2P =Q

  • P = 2Q

  • P = 4Q


67.

If x = 1 + 12 × 1! + 14 × 2! + 18 × 3! + ... and y = 1 + x21! + x42! + x63! +... Then, the value of logey is

  • e

  • e2

  • 1

  • 1e


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68.

The value of the infinite series

12 + 223! + 12 + 22 + 324! + 12 + 22 + 32 +425! + ... is

  • e

  • 5e

  • 5e6 - 12

  • 5e6


C.

5e6 - 12

Given infinite series is,

12 + 223! + 12 + 22 + 324! + 12 + 22 + 32 +425! + ...

nth term, tn12 + 22 + 32 +42 + ... + r + 12r + 2!

Now, Sn = tn = r = 1n12 + 22 + 32 + ... r + 1r + 2!                           = r = 1nr + 1r + 22n + 36r + 2!                                  n2 = nn + 12n + 16= r = 1n2r + 36 r! = 16r = 1n2r - 1! + 3r!= 1621 + 31! + 21! + 32! + 22! + 33! +...= 1621 + 21! + 22! + ...  + 311! + 12! + ... = 1621 + 11! + 12! + ... + 311! + 12! + ...= 162e + 3e - 1= 162e +3e - 3= 165e - 3= 5e6 - 12


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69.

The sum of the series 1 + 12C1n + 13C2n + ... + 1n + 1Cnn is equal to

  • 2n + 1 - 1n + 1

  • 32n - 12n

  • 2n + 1n + 1

  • 2n + 12n


70.

The value of r = 21 + 2 + ... + r - 1r!

  • e

  • 2e

  • e2

  • 3e2


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