Singapore Mental Math (3) – adding a series of numbers in a pattern

See previous posts (here and here) if you are new to this.  Here is the Week 4 strategy.

Good thing: lots of practise of following an algorithm, and of multiplying by 5, 7 or 9 mentally.

Bad thing: while some questions have 5 numbers (‘addends’!), some 7 and others 9, all of the series go up by 2 each time (the method actually works for any linear sequence).

This is a missed opportunity if only used to answer these questions in the way stated.

Why does this method work?  This is a nice example of where algebra isn’t enough by itself: this is so much easier to see and to understand if you choose you algebra in a particular way.  Make the middle number in the series be n, and this is now straightforward.  (To be clear, it still works if you make the first number n, but this is more difficult to generalise from.)

With a 5-number series that increases by 2 we have:  (n-4) + (n-2) + n + (n+2) + (n+4)

When we sum this we get 5n, which is 5 times the middle number (as per the algorithm).  But we can see why this works: there is some symmetry in that the number to the left of n is 2 below n while the number to the right of n is 2 above n, so these cancel each other out.  If you have additional numbers then they will also cancel each other out, leaving only extra ns.

Can we do this if the series increases by 3?  Yes – that gives us: (n-6) + (n-3) + n + (n+3) + (n+6)

What if the series increases by a?  (n-3a) + (n-2a) + (n-a) + n + (n+a) + (n+2a) + (n+3a)

What if there is an even number of terms?
This still works (although the 'middle' is no longer an integer.

We can also use a visual method for this:

The first diagram shows 6+8+10+12+14.  In the second diagram that has been converted to 5 lots of the middle number.

This diagram also shows neatly that we are working out the mean of the numbers in the sequence (the middle number is the mean) and then multiplying by the number of terms. 

As a final different method, it would be nice to show this on a 100-square to illuminate further what is going on.