JEE Main & Advanced Mathematics Sequence & Series Question Bank Self Evaluation Test - Sequences and Series

The value of x + y + z is 15 if a, x, y, z, b are in A.P. while the value of \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] is \[\frac{5}{3}\] if a, x, y, z, b are in H.P. Then the value of a and b are

A) 2 and 8

B) 1 and 9

C) 3 and 7

D) None

Correct Answer: B

Solution :

[b] As x, y, z, are A.M. of a and b

\[\therefore \,\,\,x+y+z=3\left( \frac{a+b}{2} \right)\]

\[\therefore \,\,\,\,\,15=\frac{3}{2}(a+b)\Rightarrow a+b=10\] ??. (1)

Again \[\frac{1}{x},\frac{1}{y},\frac{1}{z}\] are A.M. of \[\frac{1}{a}\]and \[\frac{1}{b}\]

\[\therefore \,\,\,\,\,\,\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{2}\,\left( \frac{1}{a}+\frac{1}{b} \right)\]

\[\therefore \,\,\,\,\,\,\frac{5}{3}=\frac{3}{2}.\frac{a+b}{ab}\]

\[\Rightarrow \,\,\,\,\,\,\,\frac{10}{9}=\frac{10}{ab}\Rightarrow ab=9\] ??.. (2)

Solving (1) and (2), we get

\[a=9,\,\,b=1,\,\,9\]