Tech bytes for tea?

Yesterday, I posted the first three numbers of a number series and left the next three numbers blank. Here’s the answer to yesterday’s teaser.

The completed sequence is:

3, 7, -4, -30, -71, -127

Do the numbers seem unrelated to each other? The formula linking these numbers is:
“ y = -15x² + 53x – 32 ”

Try it, substitute x=1, 2 and 3 in the above equation and you should get the first three numbers (3, 7, -4) provided in the original question. Put x=4, 5 and 6 to get the next three numbers (-30, -71, -127) in the “sequence”.

If you plot the numbers on a graph, you’ll get a standard U-shaped curve (upside down in this instance) governed by the above quardratic equation.

How does one derive the equation from the first three numbers?

Here are the details for those maths geeks out there:
(more…)

Can’t figure out the next few numbers to fill in to continue a number sequence? Here’s how to continue a number sequence given any numbers in the sequence.

Fill in the blanks in the following sequence:

“3, 7, 11, _ , _ , _”

My 10-year-old came back from school one day and proclaimed that one could use an Excel spreadsheet to complete an arithmetic progression like the one above. That’s an interesting application for a spreadsheet program, I thought.

But it also reminded me of my own school days when we were wrestling with our own number progressions.

There was among our group of secondary school boys this fellow by the name of “The Maths Genius” who devoured complex maths problems for tea.

I recall one day we were poring over some maths test papers that we’d just gotten back from the teacher and were screaming bloody murder over how many of our answers for the completion of number sequences were marked wrong. You see, the relationship between consecutive numbers in those sequences weren’t as straightforward as adding the same number to the previous number.

Along came Mr Maths Genius, calmly pointing out that we should’ve gotten full marks for those questions, since theoretically, one could make a number sequence from a list of ANY numbers! Hrrmmph!

He proceeded to throw us a starting sequence of 2, -5 and 7, from which to extrapolate the next three numbers.

No way could that be part of a number sequence, we howled. But he assured us that there was a maths formula for which if you plug in 1, 2, 3 (X-values representing the position of the number in the sequence), you will get the values 2, -5 and 7 as the numbers occupying positions 1, 2 and 3 of the sequence.

That formula could then be used to continue the sequence by plugging in 4, 5, 6 etc as X-values into the formula.

A simple sequence “2, 4, 6, 8, …” would have a formula “y=2x”.
The earlier sequence “3, 7, 11, …” would have a formula “y=4x-1”.
A more complex sequence “1, 4, 9, …” would have a formula “y=x²“.

When he saw our look of disbelief, he told us the formula for this sequence happened to be “y=19x²-71x+56″. So we plugged 1, 2 and 3 into the formula and guess what? We got ourselves the starting sequence of 2, -5 and 7! It was a matter of plugging 4, 5 and 6 into that same formula to get 38, 88 and 157 as the next three numbers in that sequence.

Maths Genius then informed us that starting from any 3 or more numbers, one could derive a formula similar to the above that allowed the sequence to be continued by plugging in the subsequent X-values.

Here’s the cheeky bit, it also means that given any three starting numbers, we can throw in any three arbitrary numbers for the next three blanks and then justify our answers by using all six numbers to derive the corresponding formula. So much for those IQ tests.

Let’s try one for tea.

Fill in the blanks in the following sequence:

“3, 7, -4, _ , _ , _” (no, that’s not a typo)

Answers will be provided tomorrow.