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In a sequence of 700 integers, each term after the first two terms is
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30 Aug 2022, 18:48

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Question Stats:

In a sequence of 700 integers, each term after the first two terms is the sum of the two preceding terms.

Which of the following could be the number of odd integers in the sequence?

Indicate all such numbers.

A. 0

B. 2

C. 233

D. 234

E. 350

F. 466

G. 467

H. 698

Which of the following could be the number of odd integers in the sequence?

Indicate all such numbers.

A. 0

B. 2

C. 233

D. 234

E. 350

F. 466

G. 467

H. 698

Re: In a sequence of 700 integers, each term after the first two terms is
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22 Sep 2022, 07:02

3

Say the first number in the sequence is a1 and the second a2. We can think of these two in terms of Odd and Even number and list the choices.

Case I

a1 and a2 can both be even.

Since subsequent numbers are found by additions, all numbers are even. Therefore, there will no odd numbers. Choice A.

Case II

a1 can be odd and a2 even.

This leads to a3 to be odd, a4 odd since

a3 = a1 + a2

Odd + Even = Odd

a4 = Odd + Even = Odd

Continuing this way,

a5 = Even

a6 = Odd

a7 = Odd

a8 = Even

..

Notice the pattern is: (Odd, Even, Odd), (Odd, Even, Odd), (Odd, Even, Odd)

The pattern is a repeating sequence of Odd, Even, Odd.

The number of this sequence is 700/3 = 233 1/3. Which is 699 + 1.

First 699 numbers will have 2/3 Odds. This equals: 699*2/3 = 466. The 700th number is also an Odd.

Therefore, the total is 467. Choice G.

Case III

a1 can be even and a2 odd.

The pattern now is :

a1 Even,

a2 Odd,

a3 Odd,

a4 Even,

a5 Odd

a6 Odd

a7 Even

Notice the the pattern (Even, Odd, Odd) repeats 233 times. And there remains the 700th number. The 700th number now is an Even number.

Therefore, total number of odds now is 466. Choice F.

Case IV

Both a1 and a2 are Odd.

The sequence is:

a1 Odd

a2 Odd

a3 Even

a4 Odd

a5 Odd

a6 Even

The sequence has like before 466 odd numbers. The 700th number is an odd number. So, total number of odds is 467. Choice G.

Case I

a1 and a2 can both be even.

Since subsequent numbers are found by additions, all numbers are even. Therefore, there will no odd numbers. Choice A.

Case II

a1 can be odd and a2 even.

This leads to a3 to be odd, a4 odd since

a3 = a1 + a2

Odd + Even = Odd

a4 = Odd + Even = Odd

Continuing this way,

a5 = Even

a6 = Odd

a7 = Odd

a8 = Even

..

Notice the pattern is: (Odd, Even, Odd), (Odd, Even, Odd), (Odd, Even, Odd)

The pattern is a repeating sequence of Odd, Even, Odd.

The number of this sequence is 700/3 = 233 1/3. Which is 699 + 1.

First 699 numbers will have 2/3 Odds. This equals: 699*2/3 = 466. The 700th number is also an Odd.

Therefore, the total is 467. Choice G.

Case III

a1 can be even and a2 odd.

The pattern now is :

a1 Even,

a2 Odd,

a3 Odd,

a4 Even,

a5 Odd

a6 Odd

a7 Even

Notice the the pattern (Even, Odd, Odd) repeats 233 times. And there remains the 700th number. The 700th number now is an Even number.

Therefore, total number of odds now is 466. Choice F.

Case IV

Both a1 and a2 are Odd.

The sequence is:

a1 Odd

a2 Odd

a3 Even

a4 Odd

a5 Odd

a6 Even

The sequence has like before 466 odd numbers. The 700th number is an odd number. So, total number of odds is 467. Choice G.

In a sequence of 700 integers, each term after the first two terms is
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03 Aug 2023, 01:25

oops my bad

gmatclubot

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