Sequences and Series
Find all possible integers whose geometric mean is 16.
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Assertion :For n∈N, (n!)3 <nn(2n+1)2n Reason :n>6,(3n)n<n!(2n)n
Insert x arithmetic means between x2 and 1.
Find two numbers whose arithmetic mean is 34 and the geometric mean is 16.
Assertion :STATEMENT -1 : If x,y,z are the sides of a triangle such that x+y+z=1, then [32x−1+2y−1+2z−1]≥((2x−1)(2y−1)(2z−1))1/3 Reason :STATEMENT-2 : For positive numbers their A.M., G.M. and H.M. satisfy the relation A.M.>G.M.>H.M.
If A1,A2 be two arithmetic means between 31 and 241, then their value are
If m arithmetic means are inserted between 1 and 31 so that the ratio of the 7th and (m−1)th means is 5:9, then the value of m is
The maximum sum of the series 20+1931+1832+⋯ is
(x−4) is geometric mean of (x−5) and (x−2) find x.