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### Sequence Question from IQ Test

Date: 06/28/2007 at 04:07:21
From: Rich
Subject: Whats the answer to this!!!

Find the missing number in this sequence: 11 > ? > 1045 > 10445.

I have no workings, it was for an IQ test and it was one of the only
ones I got wrong.  Frustratingly, they did not show us the answers
after.

Date: 06/28/2007 at 05:08:33
From: Doctor Ricky
Subject: Re: Whats the answer to this!!!

Hi Rich,

Thanks for writing Dr. Math!

Sequences like these, especially those given on IQ tests, tend to
test your ability to notice multiple patterns at once.  Generally IQ
tests are multiple-choice, but you may not remember the choices.

The answer for this question is 105, and it is a matter of noticing a
pattern.  First, notice that 10445 is almost the same as 1045*10.

1045*10 = 10450

To get to 10445, we simply have to subtract 5 from this, which gives
us the formula that to get the next term, which I will write as
n_(k+1), we need to multiply each term n_k by 10 and then subtract 5.

i.e.  n_(k+1) = 10*(n_k) - 5

The way we know this will be the formula is by checking to make sure
it works using the second (missing) term.  That means that if this
formula works, then:

? = 10*11 - 5 = 105

If THIS is really the missing value, then we can plug it into the
formula to confirm that it gives us the third term as being 1045.

10*105 - 5 = 1050 - 5 = 1045

Since this is true, it must be the correct term.

This is confirmed also by various other criteria we can identify
with the terms we DO have and the patterns they give us, as follows:

The first thing I noticed is that the number of digits in each term is
larger than the previous term by 1.  In other words, the fourth term
has 5 digits, the third term has 4 digits, so the second term must
have 3 digits to follow this pattern.

Another thing I noticed was the sum of the digits.  The digits of
the first term add to give 2, the digits of the third term add to
give 10, and the digits of the fourth term add to give 14.  This
pattern forms an arithmetic sequence starting at 2 and adding 4 to
each consecutive term.  This means that the sum of the digits of the
second term must be 6 since 2 + 4 = 6.

Also, each term begins with a 1, so the second term will also begin
with a 1.

Notice that all of these criteria fit our answer of 105.  The
majority of mathematics or sequences used on IQ tests typically just
involve patterns and your ability to "see" the patterns.

Let me know if you have any more questions.

- Doctor Ricky, The Math Forum
http://mathforum.org/dr.math/
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