Problem Solving

In a certain sequence, each term beyond the second term is equal to the average of the previous two terms. If a1 and a3 are positive integers, which of the following is NOT a possible value of a5

A.-9/4
B.0
C.9/4
D.75/8
E.41/2

talaangoshtari wrote:In a certain sequence, each term beyond the second term is equal to the average of the previous two terms. If a1 and a3 are positive integers, which of the following is NOT a possible value of a5

A.-9/4
B.0
C.9/4
D.75/8
E.41/2

Let the a� = x and a₃ = y, where x and y are positive integers.

Since a₃ = (1/2)(a� + a₂), we get:
y = (1/2)(x + aâ‚‚)
2y = x + aâ‚‚
aâ‚‚ = 2y - x.

Since a₄ = (1/2)(a₂ + a₃), we get:
aâ‚„ = (1/2)(2y - x + y) = (3y - x)/2.

Since a₅ = (1/2)(a₃ + a₄), we get:
aâ‚… = (1/2)[ y + (3y - x)/2 ]

= y/2 + (3y - x)/4

= (5y - x)/4

= (10y - 2x)/8

= (even integer - even integer)/8

= (even integer)/8.

Since aâ‚… = (even integer)/8, it is not possible that aâ‚… = 75/8.

The correct answer is D.

Hi,

Good Explanation But How we can use (5y-x)/4 to deduce the answer.

Going for (10y-2x)/8 doesn't strikes to the mind at any chance

Thanks

nikhilgmat31 wrote:Hi,

Good Explanation But How we can use (5y-x)/4 to deduce the answer.

Going for (10y-2x)/8 doesn't strikes to the mind at any chance

Thanks

Question stem: Which of the following is NOT a possible value of aâ‚…?
When the question stem includes the phrase in red, the goal of the test-writers is to make us test all five answer choices.
For this reason, the correct answer choice is often D or E.
Set (5y-x)/4 equal to each of the answer choices, starting at the BOTTOM.

E: 41/2
(5y-x)/4 = 41/2
(5y-x)/4 = 82/4
5y-x = 82.
Many combinations of x and y will satisfy this equation.
Since the resulting equation is valid, 41/2 is a possible value for (5y-x)/4.

D: 75/8
(5y-x)/4 = 75/8
(10y-2x)/8 = 75/8
10y - 2x = 75
5y - x = 37.5.
Since x and y are INTEGERS, the resulting equation is not valid.
Implication:
75/8 is NOT a possible value for (5y-x)/4.

The correct answer is D.