Absolutely ...

Algebra Level 3

$n$ is a positive integer
Define $f_{n}(x)=|||...|||x-n|-(n-1)|-(n-2)|...-3|-2|-1|$
If $f_{n}(x)=0$ has 2017 distinct real roots,find the sum of all possible $n$ s
(For example, $f_{5}(x)=|||||x-5|-4|-3|-2|-1|$ )

The answer is 64.

1 solution

C Anshul
Jul 16, 2018

$Hints:$

I just observe for $n=1$ real roots are $1$

$n=2$ real roots are $2$

$n=3$ real roots are $4$

$n=4$ real roots are $7$

$n=5$ real roots are $11$

So there is sequence in which common difference is in $A.P$ .

Rest is easy

$n$ th term is sum of first term and all the common differences i.e. $1+(1+2+3+........+(n-1))=$

$1+\frac{n^{2}-n}{2}$

Equating this to 2017 to get $n=64,-63$

Given n is positive so $n=64$ .

Absolutely ...

The answer is 64.

1 solution

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