PrepBharat

1. Let \[F_{0}=1,F_{1}=1\]   and \[F_{n+1}=F_{n}+F_{n-1}\forall n\geq 1\]
Sum of the series \[\sum_{n=1}^{\infty}\frac{F_{n}}{F_{n-1}F_{n+1}}\]    is
a) 1
b) 2
c) 1/2
d) 2/3

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2. If the sum to n terms of an A.P. is \[3n^{2}+5n\]  , while \[T_{m}=164\]  , then value of m is
a) 25
b) 26
c) 27
d) 28

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3. If \[G_{1}\] and \[G_{2}\] are two geometric means and A is the arithmetic mean inserted between two positive numbers a and b then the value of \[\frac{G_1^2}{G_{2}}+\frac{G_2^2}{G_{1}}\]   is
a) A
b) 2A
c) A/2
d) 3A/2

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4. If \[A_{1},A_{2}\]  be two arithmetic means and \[G_{1},G_{2}\]  be two geometric means between two positive numbers a and b, then \[\frac{A_{1}+A_{2}}{G_{1}G_{2}}\]    is equal to
a) \[\frac{a}{b}+\frac{b}{a}\]
b) \[\frac{1}{a}+\frac{1}{b}\]
c) \[\sqrt{\frac{a}{b}+\frac{b}{a}}\]
d) \[\frac{ab}{a+b}\]

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5. Suppose for each \[n\epsilon N\] .
\[\left(1^{2}-a_{1}\right)\left(2^{2}-a_{2}\right)+....+\left(n^{2}-a_{n}\right)=\frac{1}{3}n\left(n^{2}-1\right)\]
then \[a_{n}\] equals
a) n
b) n-1
c) n+1
d) 2n

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6. Let \[A_{1},A_{2}\]  be two arithmetic means, \[G_{1},G_{2}\]   be two geometric means, and H₁, H₂ be two harmonic means between two positive numbers a and b. The value of \[\frac{G_{1}G_{2}}{H_{1}H_{2}}.\frac{H_{1}+H_{2}}{A_{1}+A_{2}}\]     is
a) 1/2
b) 1
c) 3/2
d) 2

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7. Sum of the series S = (n) (n) + (n – 1) (n + 1) + (n – 2) (n + 2) + ... + 1(2n + 1) is
a) \[n^{3}\]
b) \[\frac{1}{6}n\left(n+1\right)\left(n+2\right)\]
c) \[\frac{1}{3}n^{3}-n^{2}\]
d) none of these

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8. Let \[a,d \epsilon \left(0,\infty\right)\]   and \[a_{r}=a+\left(r-1\right)d\forall r\epsilon N\]
If \[S_{k}=\sum_{i=1}^{k}\frac{1}{a_{i}}\]     then \[\sum_{k=1}^{n}\frac{k}{S_{k}}\]     cannot exceed
a) \[\frac{n}{4}\left(3a_{1}+a_{n}\right)\]
b) \[n\left(3a_{1}+a_{n}\right)\]
c) \[\sum_{k=1}^{n}a_{k}\]
d) All of the Above

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9. Let \[S_{n}=\frac{3}{2}.\frac{1}{1^{2}}+\frac{5}{2}.\frac{1+2}{1^{2}+2^{2}}+\frac{7}{2}.\frac{1+2+3}{1^{2}+2^{2}+3^{2}}+....\]
upto n terms, then \[S_{n}\] cannot exceed
a) 4n
b) 2n
c) 3n
d) All of the Above

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10. Let x and y be two positive real numbers. Let P be the rth mean when n arithmetic means are inserted between x and y and Q be the rth harmonic mean between x and y when n harmonic means are inserted between x and y, then \[\frac{P}{x}+\frac{y}{Q}\]   is independent of
a) n
b) r
c) both n,r
d) All of the Above

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