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In the non-decreasing sequence of odd in...

In the non-decreasing sequence of odd integers `(a_(1),a_(2),a_(3),....)={1,3,3,3,5,5,5,5,5....}` each positive odd integer k appears k times. It is a fact that there are integers b,c and d such that for all positive integer `n,a_(n)=b[sqrt(n+c)]+d` (where [.] denotes greatest integer function). The possible value of `(c+d)/(2b)` is

A

(a)0

B

(b)1

C

(c)2

D

(d)4

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To solve the problem step by step, we will analyze the given sequence of odd integers and derive the values of \( b \), \( c \), and \( d \) as described in the question. ### Step 1: Understanding the Sequence The sequence given is \( (a_1, a_2, a_3, \ldots) = \{ 1, 3, 3, 5, 5, 5, 5, 5, \ldots \} \). Each positive odd integer \( k \) appears \( k \) times. ### Step 2: Finding the General Form We know that the sequence can be expressed in the form: \[ a_n = b \left\lfloor \sqrt{n + c} \right\rfloor + d \] where \( \lfloor . \rfloor \) denotes the greatest integer function. ### Step 3: Analyzing the Values of \( a_n \) - For \( n = 1 \): \( a_1 = 1 \) - For \( n = 2 \): \( a_2 = 3 \) - For \( n = 3 \): \( a_3 = 3 \) - For \( n = 4 \): \( a_4 = 5 \) - For \( n = 5 \): \( a_5 = 5 \) - For \( n = 6 \): \( a_6 = 5 \) - For \( n = 7 \): \( a_7 = 5 \) - For \( n = 8 \): \( a_8 = 5 \) - For \( n = 9 \): \( a_9 = 7 \) ### Step 4: Setting Up Equations From the values of \( a_n \), we can set up equations. Let's start with \( n = 1 \): \[ a_1 = b \left\lfloor \sqrt{1 + c} \right\rfloor + d = 1 \] This simplifies to: \[ b \left\lfloor \sqrt{1 + c} \right\rfloor + d = 1 \quad (1) \] Next, for \( n = 2 \): \[ a_2 = b \left\lfloor \sqrt{2 + c} \right\rfloor + d = 3 \] This gives us: \[ b \left\lfloor \sqrt{2 + c} \right\rfloor + d = 3 \quad (2) \] ### Step 5: Solving the Equations From equation (1): Assuming \( \left\lfloor \sqrt{1 + c} \right\rfloor = 0 \) or \( 1 \): 1. If \( \left\lfloor \sqrt{1 + c} \right\rfloor = 0 \): - \( \sqrt{1 + c} < 1 \) implies \( c < 0 \). - Then \( d = 1 \) leads to \( b \cdot 0 + 1 = 1 \) which is valid. 2. If \( \left\lfloor \sqrt{1 + c} \right\rfloor = 1 \): - \( 1 \leq \sqrt{1 + c} < 2 \) implies \( 0 \leq c < 3 \). - Then \( b + d = 1 \). From equation (2): Assuming \( \left\lfloor \sqrt{2 + c} \right\rfloor = 1 \) or \( 2 \): 1. If \( \left\lfloor \sqrt{2 + c} \right\rfloor = 1 \): - \( 1 \leq \sqrt{2 + c} < 2 \) implies \( -1 < c < 2 \). - Then \( b + d = 3 \). ### Step 6: Finding Values of \( b, c, d \) From the analysis, we can deduce: - Set \( b = 2 \), \( d = 1 \), and \( c = -1 \). ### Step 7: Calculating \( \frac{c + d}{2b} \) Now, we compute: \[ c + d = -1 + 1 = 0 \] \[ 2b = 2 \times 2 = 4 \] Thus, \[ \frac{c + d}{2b} = \frac{0}{4} = 0 \] ### Final Answer The possible value of \( \frac{c + d}{2b} \) is \( \boxed{0} \).
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ARIHANT MATHS ENGLISH-MONOTONICITY MAXIMA AND MINIMA-Exercise (Passage Based Questions)
  1. In the non-decreasing sequence of odd integers (a(1),a(2),a(3),....)={...

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  2. In the non-decreasing sequence of odd integers (a(1),a(2),a(3),....)={...

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  3. In the non-decreasing sequence of odd integers (a(1),a(2),a(3),....)={...

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  4. Let g(x)=a(0)+a(1)x+a(2)x^(2)+a(3)x^(3) " and " f(x)=sqrt(g(x)), f(x) ...

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  5. Let g(x)=a(0)+a(1)x+a(2)x^(2)+a(3)x^(3) " and " f(x)=sqrt(g(x)), f(x) ...

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  6. Let g(x)=a(0)+a(1)x+a(2)x^(2)+a(3)x^(3) " and " f(x)=sqrt(g(x)), f(x) ...

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  7. f: D->R, f(x) = (x^2 +bx+c)/(x^2+b1 x+c1) wherealpha,beta are the root...

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  8. f: D->R, f(x) = (x^2 +bx+c)/(x^2+b1 x+c1) wherealpha,beta are the root...

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  9. f: D->R, f(x) = (x^2 +bx+c)/(x^2+b1 x+c1) wherealpha,beta are the root...

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  10. f: D->R, f(x) = (x^2 +bx+c)/(x^2+b1 x+c1) wherealpha,beta are the root...

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  11. f: D->R, f(x) = (x^2 +bx+c)/(x^2+b1 x+c1) wherealpha,beta are the root...

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  12. consider the function f(x)=(x^(2))/(x^(2)-1) The interval in which f...

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  13. consider the function f(x)=(x^(2))/(x^(2)-1) If f is defined from R-...

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  14. consider the function f(x)=(x^(2))/(x^(2)-1) f has

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  15. Let f(x)=e^((P+1)x)-e^(x) for real number Pgt0, then The value of x=...

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  16. Let f(x)=e^((P+1)x)-e^(x) for real number Pgt0, then Use the fact t...

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  17. Let f(x)=e^((P+1)x)-e^(x) for real number Pgt0, then Let g(t)=int(t)...

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  18. Consider f, g and h be three real valued function defined on R. Let f(...

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  19. Consider f, g and h be three real valued function defined on R. Let ...

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  20. Consider f, g and h be three real valued function defined on R. Let f(...

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