Population of cattle in a farm increases so that difference between population in year n+2 and that

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4-2.Quadratic Equations and Inequations

normal

The population of cattle in a farm increases so that the difference between the population in year $n+2$ and that in year $n$ is proportional to the population in year $n+1$. If the populations in years $2010, 2011$ and $2013$ were $39,60$ and $123$,respectively, then the population in $2012$ was

$81$

$84$

$87$

$90$

(KVPY-2014)

Solution

(b)

Given,

Population in year $2010, 2011$ and $2013$ were $39,60$ and $123$ respectively.

According to problems,

The population of cattle in farm increases such that difference between in year $n+2$ and that in year $n$ is proportional to the year $n+1$

$\because \quad(n+2)-(n)=k(n+1)$

$\because$ Let population in year $2012=x$

$Year$	$Population$
$2010$	$39$
$2011$	$60$
$2012$	$x$
$2013$	$123$

From Eqs.$(i)$ and $(ii)$, we get

$\frac{x-39}{60}=\frac{123-60}{x}$

$\Rightarrow x^2-39 x-3780=0$

$\Rightarrow (x-84)(x+40)=0$

$\therefore x=84$

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