Data Sufficiency (DS) | If f(x) = f(x - 1) + f(x - 2) for all x > 2; is f(5) a perfect square

\( f(5) =f(4)+f(3).\)..(i)

\( f(4) =f(3)+f(2)...(ii)\)

\( f(3) = f(2)+f(1)...(iii)\)

\( f(5) = f(2)+ f(1) +f(2) +f(2)+f(1)...\) putting (ii) and (iii) in (i)

Thus\( f(5) = 3f(2) +2f(1)...(iv)\)

Thus if we can find \(f(2)\) mand\(f(1)\) we can answer thestem.


(1) f(2) = 2f(1)

Using\((iv)\) we have\( 6f(1)+2f(1) =8f(1)\)

However we do not know\(f(1)\)

INSUFF.

(2) The first two terms of the series, f(1) and f(2) are consecutive evennumbers.

We could have\(f(1)\) and\(f(2)\) with many possibilities suchas:

\(2\) and\( 4 \)
\(0\) and\( 2 \)
\( 0\) and\( -2\)
\(-2\) and\( -4\)
etc.

As such we will get many values for\( f(5) = 3f(2)+2f(1)\)

INSUFF.

1+2

\( f(1)=-2\) and\( f(2)=-4\) or:
\( f(1)=\hspace{1mm}2\hspace{1mm}\) and\( \hspace{3mm} f(2)=\hspace{1mm}4\)

Thus using eqn.(iv) we can get\( f(5)=16\) or\( f(5)= -16\)

Hence we can get both aYES and a NO.

IMO ans should be E
...

Statistics : Posted by stne • on 26 Oct 2023, 22:46 • Replies 1 • Views 120

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