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Each term, starting from the third term, of a sequence

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Each term, starting from the third term, of a sequence [#permalink]

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New post 17 Jul 2017, 22:26
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Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14

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Each term, starting from the third term, of a sequence [#permalink]

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New post 17 Jul 2017, 23:09
TimeTraveller wrote:
Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14


It should be B.

If first two terms are Odd, third would be Even, 4th would be E + O = Odd, so sequence becomes.

O O E O

fifth would again be Odd because E + O = O

O O E O O

Now 6th term would be Even because 4th and 5th are both odd and this sequence repeats.

O O E O O E O O E

If this sequence goes on 33 times, terms would be 99 and last term would be even but we have trimmed the last Even term out.

so we have 33 x 2 odd and 32 Even terms.

66:32 = 33:16
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Re: Each term, starting from the third term, of a sequence [#permalink]

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New post 18 Jul 2017, 03:30
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TimeTraveller wrote:
Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14


Use Pattern Recognition:

First two are odd terms.

Odd + Odd = Even
So third term is even. We have Odd, Odd, Even

Odd + Even = Odd
So fourth term is Odd. We have Odd, Odd, Even, Odd

Even + Odd = Odd
So fifth term is Odd. We have Odd, Odd, Even, Odd, Odd

Now, we again have 2 odd terms to the pattern will continue.

Odd, Odd, Even, Odd, Odd, Even, Odd, Odd, Even... and so on
In every 3 terms, 2 are Odd and 1 is Even. So in 99 terms, 66 would be Odd and 33 would be Even,the last one being Even.
If we have only 98 terms, we would have 66 Odd and 32 Even.

Required Ratio = 66/32 = 33/16

Answer (B)
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Re: Each term, starting from the third term, of a sequence [#permalink]

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New post 20 Jul 2017, 16:48
TimeTraveller wrote:
Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14


Since the first two terms are odd, we have:

1st term: odd

2nd term: odd

3rd term: odd + odd = even

4th term: even + odd = odd

5th term: even + odd = odd

6th term: odd + odd = even

7th term: even + odd = odd

8th term: odd + even = odd

9th term: odd + odd = even

We see that each term that is a multiple of 3 is even and all other terms are odd.

From 1 to 98, there are (96 - 3)/3 + 1 = 32 multiples of 3, and thus there are 32 even terms. So, there are 98 - 32 = 66 odd terms. So, the ratio of odd-valued terms to even-valued terms is 66/32 = 33/16.

Answer: B
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Re: Each term, starting from the third term, of a sequence [#permalink]

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New post 20 Jul 2017, 21:15
TimeTraveller wrote:
Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14


for each 3 term cycle, the sequence is odd-odd-even
98/3=32 2/3 cycles
32*1=32 even terms
32*2+2=66 odd terms
66:32=33:16
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Re: Each term, starting from the third term, of a sequence [#permalink]

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New post 08 Sep 2017, 12:13
VeritasPrepKarishma wrote:
TimeTraveller wrote:
Each term, starting from the third term, of a sequence, which has 98 terms, is the sum of the two preceding terms. If the first two terms of the sequence are odd numbers, then what is the ratio of the number of odd valued terms to that of even valued terms?

(A) 34 : 15
(B) 33 : 16
(C) 30 : 19
(D) 29 : 20
(E) 35 : 14


Use Pattern Recognition:

First two are odd terms.

Odd + Odd = Even
So third term is even. We have Odd, Odd, Even

Odd + Even = Odd
So fourth term is Odd. We have Odd, Odd, Even, Odd

Even + Odd = Odd
So fifth term is Odd. We have Odd, Odd, Even, Odd, Odd

Now, we again have 2 odd terms to the pattern will continue.

Odd, Odd, Even, Odd, Odd, Even, Odd, Odd, Even... and so on
In every 3 terms, 2 are Odd and 1 is Even. So in 99 terms, 66 would be Odd and 33 would be Even,the last one being Even.
If we have only 98 terms, we would have 66 Odd and 32 Even.

Required Ratio = 66/32 = 33/16

Answer (B)



VeritasPrepKarishma,

Hello,

tried a different approach

98 Numbers can be divided in 14 sets each having 7 numbers. In each set we have 5 odd and 2 Even

Then number of Odd is 14*5= 70 and number of even is 2* 14= 28

Then the ratio is 70:28

Can you help me identify my error?

Thanks
Abhimanyu
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Re: Each term, starting from the third term, of a sequence   [#permalink] 08 Sep 2017, 12:13
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