Algebra 1 extra credit problems

[dpulavarthi]

In question 3, The sequence an is defined so that, for all
n ≥ 3, an is the greater of (an - 2 + 1) and
(an - 1). (If the two quantities are the same,
then an is equal to either of them.) Which of
the following values of a1 and a2 will produce
a sequence in which no value is repeated?

How do you approach this problem?

(A) a1 = -1, a2 = -1.5
(B) a1 = -1, a2 = 1
(C) a1 = 1, a2 = -1
(D) a1 = 1, a2 = 1.5
(E) a1 = 1.5, a2 = 1

[tim]

Before we help with this question, we need you to show some effort of your own. What did you try on this question? Where did you get stuck?

[dpulavarthi]

I tried to calculate the values for a1, a2, a3 but then wasnt sure of because it says that the below inequality is true only for n>=3
So is it correct if used the ineqaulity given in the question to calculate a1,a2,a3. Start with a3 work my way backwards.

But I couldn't find any pattern. Is this the correct approach?

[tim]

you won't need to calculate a1 and a2; just plug in the values from the answer choices, and then you can work forward to get a3 and so on. try that and see if it works for you..

[dpulavarthi]

Sorry. I still don't get it. Where am I plugging in the values?

Here is what I did after your hint.

a1=-1, a2=-1.5

Question says for n>=3, an-1 (n-1 in the subscript) < an
an-2 + 1 < an.

so lets say n=3,

then an-2 +1 = a1+1=0
then an=a3,
an-1=a2=-1.5

How do I find out what is repeating? The answer key says option A: 0 is repeating.

[jlucero]

I think your misunderstanding is the phrase "a(n) is the greater of [a(n - 2) + 1] and [a(n - 1)]." There's no shortcut for this type of question, other than solving for each of these and finding out which one is larger. So for answer choice (A), the first two numbers in the sequence are:

-1, -1.5, __

So what would the next number in the sequence be? a(3) would be "the greater of [a(3 - 2) + 1] and [a(3 - 1)]" For these two numbers, you would need to take [-1 + 1] and [-1.5] and find out that the first number is larger (0).

-1, -1.5, 0, __

Continue this pattern and you would find that the next number in the sequence would be [-1.5 + 1] or [0]. Since zero is larger, the next number in the sequence would be [0]. You can continue this pattern to find that the next few numbers in the sequence would be:


-1, -1.5, 0, 0, 1, 1, 2, 2...

But once you find a number that is repeated, you can really stop. So which of the five answer choices wouldn't have a number that repeats?

[kapilreadyforgmat]

So answer will be D.

A3=greater of (A1+1,A2) => (2, 1.5) => 2
A4=greater of (A2+1,A3) => (2.5, 2) => 2.5
A5=greater of (A3+1,A4) => (3,2.5)=> 3
. .
. .
. .
and so on.

Thanks
Kapil

[tim]

:)