Class 11

Math

Algebra

Sequences and Series

Let $x$ be the arithmetic mean and $y,z$ be the two geometric means between any two positive numbers. Then $xyzy_{3}+z_{3} $ = __________.

Given that $x$ is the A.M. between $a$ and $b$ and $y$, and $z$ are the G.M. $s$ between $a$ and $b$ where $a$ and $b$ are positive. Then $a,x,b$ are in A.P. So,

$x=2a+b $

$a,y,z,b$ are in G.P. So,

$y=ar$ and $z=ar_{2}$, where $r=3 cb $. Also, $yz=ab$, Now,

$xyzy_{3}+z_{3} =ab(2a+b )a_{3}r_{3}+a_{3}r_{6} $

$=ab(2a+b )a_{3}×ab +a_{3}×a_{2}b_{2} $

$=a_{2}b+ab_{2}2(a_{2}b+ab_{2}) $

$xyzy_{3}+z_{3} =2$