perplexus.info :: Sequences : Last term zero

Last term zero (Posted on 2016-01-28)

{S(0), S(1), S(2), ....., S(n) } is a sequence of real numbers such that:
S(0) = 37, S(1) = 72, ....., S(n) = 0, and:

S(k+1) = S(k-1) – 3/S(k) for k=0,1,2,...., n-1

Find n

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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solution

| Comment 1 of 3

let P(k)=S(k)S(k+1)

then we have

P(k)=S(k)*[S(k-1)-3/S(k)]

P(k)=S(k-1)S(k)-3

P(k)=P(k-1)-3

with P(0)=S(0)*S(1)=37*72=2664

thus

P(k)=2664-3k

now when P(k)=-3 then we have

-3=S(k)S(k-1)-3

S(k)S(k-1)=0

thus one of S(k) or S(k-1) is zero

so we have P(n)=-3

2664-3n=-3

3n=2667

n=889

so either S(888) or S(889) is zero

assume S(888)=0

then S(889)=S(887)-3/S(888)

which results in division by zero

thus S(889) is undefined

but if S(889) is undefined then so is P(889)=S(889)*S(890)

contradiction

thus S(888) is not zero and thus S(889) is zero

Thus n=889

Posted by Daniel on 2016-01-28 09:23:03

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