The nth term of an increasing sequence S is given by Sn = Sn-1 + Sn-2

Question

The nth term of an increasing sequence S is given by  S_n = S_{n-1} + S_{n-2}  for n > 2 and the nth term of a sequence S’ is given by  S’_n = S’_{n-1} - S’_{n-2}  for n > 2. If  S_5 = S’_5 , what is the average (arithmetic mean) of  S_2  and  S’_2 ?

(1) The difference between the fourth term and the second term of sequence S is 14.

(2) The sum of the fourth term and the second term of sequence S’ is 14.

AbhimanyuDhar · Accepted Answer

S(5)= s4+s3= s3+s2+s2+s1= S2+s1+s2+s2+s1= s2+2(s1+s2)
S'(5)=s'4-s'3=s'3-s'2-s'3= =-s'2

So -s'2= s2+2(s1+s2)
or (s2+s'2)/2= -(s1+s2)

To find : -(s1+s2) or s3.

STMT 1:
s4-s2= s3+s2-s2= s3= 14

We know that s3=s1+s2 and that is required to find

STMT 2: 
s'4+s'2=14
s'3-s'2+s'2=14
s'3=14
No use of this value as it does not help us to reach the value of s3 or s1+s2

Hence A

amanvermagmat · Answer

Let the first 5 terms of sequence S be represented as: S1, S2, S3, S4, S5
Here S3 = S2+S1, S4 = S3+S2, S5 = S4+S3 ..

Let the first 5 terms of sequence S' be represented as: S1', S2', S3', S4', S5'
Here S4' = S3'-S2', S5' = S4'-S3'

Now lets look at the statements: (What I will try to do is write as many terms as possible in terms of either S2 or S2' because our objective is to calculate mean of S2 & S2', which we will get once we have the sum S2+S2')

Statement 1. S4-S2 = 14, Or S4 = 14+S2, 
But S4 = S3+S2.. this means S3 = 14
Thus S5 = S4+S3 = 14+S2 + 14 = 28+S2
So S5 can be written in terms of S2 as '28+S2'.

We are given that S5 = S5', so S5' = 28+S2
Now, S5' = S4' - S3' and S4' = (S3'-S2') - S3' = -S2'

See, S5' can be written as (28+S2) and it can also be written as '-S2' . Equating them both:
28+S2 = -S2' Or S2+S2' = -28 
We have their sum, so their average = -28/2.  Sufficient .

Statement 2. S4' + S2' = 14. Now, S4' = S3'-S2' or S4'+S2' = S3'

But S4' + S2' = 14, So S3'=14.... Thus we can say S4' = S3'-S2' = 14-S2', and S5' will become:
S5' = S4'-S3' = 14-S2' - 14 = -S2'

We are given that S5 = S5' so S5 = -S2' or S4+S3 = -S2'
Replacing S4 with S3+S2, we have S3+S2 + S3 = -S2' or S2+S2' = -2S3

We have their sum in terms of variable S3, so mean cannot be calculated.  Insufficient .

Hence  A answer

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The nth term of an increasing sequence S is given by Sn = Sn-1 + Sn-2

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