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Expert: - 6/4/2009


Question
QUESTION: Hi... I am preparing for my SAT in three days and I have been out of highschool for nearly 4 years. ( I spent the four years in the military.)  So i am a little rusty.  Can you please explain the answer to this problem???

3,5,-5.....
The first term in the sequence of numbers shown above is 3.  Each even-numbered term is 2 more than the previous term and each odd-numbered term, after the first, is -1 times the previous term.  For example, The second term is 3+2, and the third term is (-1) x 5.  What is the 55th term?

I thought the answer was 3.... However the answer key says the correct answer is -5.  How did they get this??!  I'm going crazy!!!

ANSWER: The 1st term is 3.
The 2nd term is 5.
The 3rd term is -2nd term = -5.
The 4th term is -5+2 = 3, which is back at the first term.
This means that the 5th term is 5.
This means that the 6th term is -5.
This means that the 7th term is back to 3 again.

This creates a repeat every 3rd term.

This means that every term of 1+3n is 3, 2+3n is 5, and 3+3n is -5.

Note the qeustion asks for the 55th term.  55 = 3*18 + 1,
so that term would be the same as the 1st term.

The first term was 3, so the 55th term is also 3.


---------- FOLLOW-UP ----------

QUESTION: I see where your thinking goes, but if the first is 3, second is 5, and third is -5.... when you add 2 to -5 you would get -3 not positive 3.... so that would change everything.... if it is -3, the next number would be -1, and third 1..... that would then be 1,3,-3.... so and so fourth... what am i doing wrong?  The correct answer listed is -5 in my SAT study guide.... HELP!


Answer
Sorry about that, so here is the right answer.

The 1st number is 3.  1st is odd, so add 2.  
This gives 5, which is the next.

The 2nd number is 5.  2nd is even, so take negative.  
This gives -5, which is the next.

The 3rd number is -5.  3rd is odd, so add 2.
This gives -3, which is the next.

The 4th number is -3.  4th is even, so take negative.  
This is the next, which is 3.

The 5th number is 3.  5th is odd, so add 2.

Now 3 was the 1st and it comes back to us at the 5th.

From this I can say that if the number is 4n+1 for all positive integers n.

The number we are given is 55th.
Since 55 = 13*4 + 3, that 55th number will be the same as the 3rd.
Note that our third number was -5.

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