The Number Devil Missing
Chapter: Sequences
By: Kenneth Braggs
Braggs
“You couldn’t possibly think that your math journey ends here?
Robert looked up startled to see his acquaintance.
“You came back!” Robert shouted excitedly.
“Of course I would but, enough of the formalities we have a somewhat new
topic to discuss.” Teplotaxl rushed.
“What do you mean by somewhat new?”
“Well this topic has been discussed by us already but now we will go into
more depth. You do remember when all of those numbers were in your room?”
Teplotaxl asked.
“Of course, but what of it?”
“Well then you should remember when we talked about the numbers going
on forever in a pattern . This topic is what we will discuss but more in depth.”
Teplotaxl explained.
“Well this should be very easy.” Robert said confidently.
“Do not claim this so fast it gets more complicated. We will begin with a
simple number set that you should recognize quite easily.”
Just as he spoke Teplotaxl commanded a sequence of blue numbers ranging
from 1 to infinity which quickly crowded the sky. Numbers stretched as far as the
eye could see and even further.
“It’s just a regular number line. That has an unnecessary amount of numbers
of course.” Robert cried.
“It will be fine just trust me. Let’s first begin with the 1 right above us.”
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“You sure love your 1’s”
“Yes of course but that is beside the point. Can you find the pattern between
these numbers?”
“I’m not dumb anyone could see that you add 1 to the previous number to
get to the next number.”
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“Just as he spoke Teplotaxl commanded a sequence of blue numbers
ranging from 1 to infinity which quickly crowded the sky. Numbers stretched as far
as the eye could see and even further.”
“Good now this pattern that you noticed is called arithmetic because there is
a constant difference between each number, the numbers between are defined as a
series and the number set is called a sequence. Now that you know that lets try a
bit trickier sequence.” said Teplotaxl.
Just as the first sequence appeared it disappeared being replaced by a
sequence whose first ten numbers were:
“This is easy too,” Robert said “Each number is multiplied by two in order
to get the next number in the sequence. Is this ever going to get any harder?”
“Be patient, but I’m glad you’re curious. This past sequence was called a
geometric sequence where there is no constant difference; however there is a
constant ratio between each number. At your request I will proceed to make this
harder.”
Again the numbers in the sky changed but this time replaced by a number set
with the first ten numbers being:
Robert looked confused and finally spit out “I don’t understand! There is
neither a constant difference nor a constant ratio!”
“Exactly.” Teplotaxl said slyly.
Robert looked more confused than before.
“There is neither a constant difference nor a constant ratio because the
sequence is neither arithmetic nor geometric. It’s simply categorized as neither but,
the equation to get a series such as this is -2n˖3 where n acts as x would in a
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function. However we will not be discussing this type of sequence much more in
the near future.”
“Well that’s dumb why would you teach me this then?”
“I needed to discus all types of sequences as well as you requested me to
make this harder. But onto more important matters, let’s discuss the formulas of
sequences.”
“Why can’t we just add, subtract, multiply, or divide?”
“Great Question!” Teplotaxl Exclaimed “Well with that logic please tell me
the 50 number of a sequence where the series is addition by 413.”
th
“No fair, that will take me forever even with a couch calculator.” Robert
whined.
“This is precisely why we use the formulas, for such sequences as these and
many others. What we were doing earlier would be referred to as the recursive
formula where an, or the number of the sequence we are looking for, equals the sum
of an-1, the previous term added to the constant ratio, d and the geometric recursive
formula is similar but instead of an a there is a g, instead of addition there is
multiplication and instead of a d there is a r for ratio.”
“That was confusing.” Said Robert
“It’s spelled out on the ground, my boy!”
As Teplotaxl spoke both of the recursive formulas appeared before them.
“It makes more sense now.” Robert said.
“But the worst part of the recursive formula is that we must know the
previous term to find the next number in the sequence. So we would have to add all
the numbers before the number of the sequence that we want, in order to find that
term.”
“Well that’s awful we would take forever to find one term.”
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“That why we use explicit formulas. We would just need to know the first
term and we could simply find any other term of the sequence. This is also a
mouthful so look at the ground and you will find this equation. We use a n, again
this time equaling the sum of the first term a1, added to d multiplied by the quantity
(n-1).”
As Teplotaxl spoke the formula once again appeared on the ground before
Robert.
“What does the (n-1) mean?” Robert asked.
“The n refers to the number input, and the -1 refers to the previous term.”
“So if we wanted to find the equation for the first sequence would substitute
1for a1, and 1for d, making the equation an=n.
“Correct” proclaimed Teplotaxl. “Now the explicit formula for a geometric
sequence will be just a little more complicated. gn, equals the product of g1,
multiplied by r, to the power of n-1.”
Robert looked down at the equation for comprehension and began his
thought:
“For this equation to find the second sequence we replace g1 with 1, and r
with 2, which yields gn=1·2n-1.”
“Exactly” Teplotaxl declared proud of Robert’s progress.
“What about the neither sequences?”
“Well those you have to find the formula for, based upon the number set.
Those sequences are not as simple as these. But, now I must go I have other
mathematicians to disturb.”
“Will you ever come back?” Robert asked.
“We’ll never know, keep dreaming.”
Just as the number devil had appeared he had vanished.
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