Given the sequence of numbers `x_(1),x_(2),x_(3),x_(4),….,x_(2005)`, `(x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+3)=(x_(3))/(x_(3)+5)=...=(x_(2005))/(x_(2005)+4009)`, the nature of the sequence is
A
`A.P.`
B
`G.P.`
C
`H.P.`
D
None of these
Text Solution
Verified by Experts
The correct Answer is:
A
`(a)` Given `(x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+3)=(x_(3))/(x_(3)+5)=...=(x_(2005))/(x_(2005)+4009)` `impliesx_(1)=(lambda)/(1-lambda)`, `x_(2)=(3lambda)/(1-lambda)`, `x_(3)=(5lambda)/(1-lambda)`,…… Hence , `x_(1),x_(2),x_(3),…..,x_(2005)` are in arithmetic progression.
Prove that for any two numbers x_(1) and x_(2) (2e^(x_1)+e^(x_2))/(3)gte(2x_(1)+x_(2))/(3)
The sequence {x_(k)} is defined by x_(k+1)=x_(k)^(2)+x_(k) and x_(1)=(1)/(2) . Then [(1)/(x_(1)+1)+(1)/(x_(2)+1)+...+(1)/(x_(100)+1)] (where [.] denotes the greatest integer function) is equal to
(3x - 9)/( (x - 1 )(x + 2 )(x^(2)+1))
Value of |{:(1+x_(1),,1+x_(1)x,,1+x_(1)x^(2)),(1+x_(2),,1+x_(2)x,,1+x_(2)x^(2)),(1+x_(3),,1+x_(3)x,,1+x_(3)x^(2)):}| depends upon
Solve, by Cramer's rule the system of equations x_(1)-x_(2)=3,2x_(1)+3x_(2)+4x_(3)=17,x_(2)+2x_(3)=7 .
Let the equation x^(5) + x^(3) + x^(2) + 2 = 0 has roots x_(1), x_(2), x_(3), x_(4) and x_(5), then find the value of (x_(1)^(2) - 1)(x_(2)^(2) - 1)(x_(3)^(2) - 1)(x_(4)^(2) - 1)(x_(5)^(2) - 1).
Consider the quantities such that x_(1),x_(2),….x_(10),-1 lex_(1),x_(2)….,x_(10)le 1 and x_(1)^(3)+x_(2)^(3)+…+x_(10)^(3)=0 , then the maximum value of x_(1)+x_(2)+….+x_(10) is
