Warm up problem 5: functions

Algebra Level 3

We define a function f(x) as f ( x ) = f ( x − 1 ) − f ( x − 2 ) f(x)=f(x-1)-f(x-2) for x>2 with f(1)=2 and f(2)=3. Find the 1 , 000 , 000 t h 1,000,000th term.

-2 1 3 -3

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Trevor Arashiro by Trevor Arashiro , 22, USA

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2 solutions

Ankush Gogoi
Sep 21, 2014

Given f(1)=2 , f(2)=3 Now f(x)=f(x-1)-f(x-2) So f(3)=f(2)-f(1)=1...similarly f(4)=f(3)-f(2)=-2...proceeding this way we get f(5)=-3 , f(6)=-1 , f(7)=2 , f(8)=3 , f(9)=1 , f(10)=-2 , f(11)=-3 f(12)=-1 , f(13)=2 ....Here we notice that after every 6 no.s the 7th no repeats the same value as that of the 1st no and so on upto the 6th no...hence taking the nos in pairs of six we get the 999996th no will have the value -1(i.e the value of the sixth no f(6)) so f(999997)=2 , f(999998)=3 , f(999999)=1 & f(1000000)=-2

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Nitin Tiwari
Sep 20, 2014

i still dnt understand completely can i have explanations please..

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Just start writing the values of f(1), f(2), f(3),....

U will notice that f(7)=f(1), f(8)=f(2),.....

The terms are repeating after every 6 terms.

Since 999,996 is a multiple of 6

So f(999,997)=f(1)

f(999,998)=f(2)

f(999,999)=f(3)

f(1,000,000)=f(4)=-2

Rohit Sachdeva - 6 years, 8 months ago

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