The sum of series  from Mathematics Sequences and Series

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 Multiple Choice QuestionsMultiple Choice Questions

31.

The sum of the first n terms of the series 1 squared space plus space 2.2 squared space plus space 3 squared plus space 2.4 squared space plus space 5 squared space plus space 2.6 squared plus... space is space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis squared over denominator 2 end fraction when n is even. When n is odd the sum is

  • fraction numerator 3 straight n space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction
  • fraction numerator straight n squared space left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction
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32.

The sum of series fraction numerator 1 over denominator 2 space factorial end fraction space plus fraction numerator 1 over denominator 4 factorial end fraction space plus fraction numerator 1 over denominator 6 space factorial end fraction space plus space.... space is

  • fraction numerator left parenthesis straight e squared minus 1 right parenthesis over denominator 2 end fraction
  • fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction
  • fraction numerator left parenthesis straight e squared minus 1 right parenthesis over denominator 2 straight e end fraction
  • fraction numerator left parenthesis straight e squared minus 1 right parenthesis over denominator 2 straight e end fraction


B.

fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction
straight e space equals space 1 space plus space fraction numerator 1 over denominator 1 factorial end fraction space plus fraction numerator 1 over denominator 2 factorial end fraction space plus fraction numerator 1 over denominator 3 factorial end fraction space plus space fraction numerator 1 over denominator 4 factorial end fraction space plus space....... infinity space.... space left parenthesis straight i right parenthesis
straight e to the power of negative 1 end exponent space equals 1 space minus space fraction numerator 1 over denominator 1 factorial end fraction space plus fraction numerator 1 over denominator 2 factorial end fraction space minus fraction numerator 1 over denominator 3 factorial end fraction space plus space fraction numerator 1 over denominator 4 factorial end fraction space minus space....... infinity space.... space left parenthesis ii right parenthesis
on space adding space equ space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
straight e space plus straight e to the power of negative 1 end exponent space space equals space 2 space plus space fraction numerator 2 over denominator 2 space factorial end fraction space plus fraction numerator 2 over denominator 4 factorial end fraction space plus........ space infinity
fraction numerator straight e squared plus 1 over denominator straight e end fraction minus 2 space equals space fraction numerator 2 over denominator 2 factorial end fraction space plus fraction numerator 2 over denominator 4 factorial end fraction space plus....... infinity
fraction numerator straight e squared space plus 1 minus 2 straight e over denominator straight e end fraction space equals space 2 space open square brackets fraction numerator 1 over denominator 2 factorial end fraction fraction numerator 1 over denominator 4 space factorial end fraction space plus..... infinity close square brackets
fraction numerator left parenthesis straight e minus 1 right parenthesis squared over denominator 2 straight e end fraction space equals space fraction numerator 1 over denominator 2 factorial end fraction space plus fraction numerator 1 over denominator 4 factorial end fraction space plus space...... infinity
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33.

Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4.
Statement-I: The numbers b1, b2, b3, b4are neither in A.P. nor in G.P. Statement-II: The numbers b1, b2, b3, b4 are in H.P.

  • Both statement-I and statement-II are true but statement-II is not the correct explanation of statement-I

  • Both statement-I and statement-II are true, and statement-II is correct explanation of Statement-I

  • Statement-I is true but statement-II is false.

  • Statement-I is true but statement-II is false.

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34.

Let a1, a2, a3, ...., a49 be in A.P. such that k = 012ak-1 = 416 and  a9 + a43 = 66. if a12 + a22 + ....... + a172 = 140m then m is equal to

  • 33

  • 66

  • 68

  • 34


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35.

Let f : R ➔ R be such that f is injective and f(x)f(y) = f(x + y) for  x, y  R. If f(x), f(y), f(z) are in G.P., then x, y, z are in

  • AP always

  • GP always

  • AP depending on the value of x, y, z

  • GP depending on the value of x, y, z


36.

In a GP series consisting of positive terms, each term is equal to the sum of next two terms. Then, the common ratio of this GP series is

  • 5

  • 5 - 12

  • 52

  • 5 +12


37.

If x is a positive real number different from 1 such that logax, logbx, logcx are in AP, then

  • b = a + c2

  • ac

  • c2 = aclogab

  • None of these


38.

If a, x are real numbers and a < 1, x < 1, then 1 + (1+ a) x + (1+ a + a2)x2 + ... is equal to

  • 11 - a1 - ax

  • 11 - a1 - x

  • 11 - x1 - ax

  • 11 + ax1 - a


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39.

If log0.3x - 1 < log0.09x - 1, then x lies in the interval

  • 2, 

  • (1, 2)

  • (- 2, - 1)

  • None of these


40.

The value of n = 113in + in + 1, i = - 1 is

  • i

  • i - 1

  • 1

  • 0


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