In this second narrative about 'What comes next?' estimate sequences, we'll reconsider how to solve more complicated arithmetic sequences, in which math skills becomes more indispensable than an ability to simply recognise and repeat patterns.
More complicated Arithmetic Sequences
In math, an arithmetic sequence or progression is a sequence of numbers in which the unlikeness in the middle of any two successive numbers in the sequence is the same. In the final example of the old article, we tested whether a sequence of numbers was a uncomplicated arithmetic progression by comparing the numerical unlikeness in the middle of adjacent pairs of numbers. It is important that adjacent pairs of numbers are chosen, as more complicated forms of arithmetic progression may be misidentified as uncomplicated arithmetic sequences if non-adjacent pairs are selected.
Consider the sequence of numbers:
2 6 4 8 6 10 _ _
If you worked out the mathematical unlikeness in the middle of the first pair and last pair of numbers, in this case both +4, you might be tempted to close that you were dealing with a uncomplicated arithmetic sequence with a base unlikeness of +4. However, the unlikeness in the middle of the first pair of numbers and the second pair of numbers is +4 and -2 respectively, so clearly there is something more complicated happening. In such cases, the most trustworthy policy of action is to work out the unlikeness in the middle of each successive pair of numbers. In this case, the differences work out to be:
+4, -2, +4, -2, +4,
At the simplest level, you could say the numerical series is generated by adding four to the first estimate to make the second number, then subtracting four from the second estimate to make the third, and so on ad infinitum. This arrival will work and allow you to complete 'What comes next?' estimate series generated in this manner. However, there is certain mathematical inelegance to the approach. To a mathematician, the series comprises two uncomplicated arithmetic progressions, each with a base unlikeness of two, which have been interleaved so that so that the series takes, alternately, numbers from one progression and then the next.
The two series when separated are:
2 4 6 8
6 8 10 12
and the next two numbers in the series 8 and 12.
Whilst in this example there are few real advantages to separating out the uncomplicated arithmetic progressions. More complicated numerical series may alternate three or more arithmetic progressions, which makes determining what math principles underlie the numerical sequence increasingly difficult.
As we're ready to move on in the next narrative to look at geometric progressions and unique estimate series, we'll close with a minute brain teaser.
What two letters come next in the following sequence?
O T T F F S _ _
This is an inspiring example of the 'what comes next?' question, in that arriving at the literal, retort will, for most people, involve elements of math, pattern matching and the spark of inspiration which we tend to reconsider innate intelligence. The question is also as likely to be answered correctly and in a inexpensive distance of time by an inspiring 5 year old as by an inspiring adult, which makes it an ideal Iq test question for measuring innate intelligence. So what is a inexpensive distance of time in which to arrive at the literal, answer? As part of an Iq test, answering correctly in less than 10 seconds would put you in genius territory, while taking nearby a minute is maybe the average. For whatever who is indeed struggling with the question, please be assured that an retort and an explanation will be forthcoming in the final 'number series' article.
