This is question 4 from the Test of Mathematics for University Admission practice paper by the University of Cambridge.

The sequence does look a bit scary at first glance, but fret not, writing out the first few terms will help you find the summation from n = 1 to 39.

Now I did this in my head, but if you’d like, you can grab a pen and a piece of paper, and give this problem a go.

When you are ready, keep reading for the solution!

Solution

Let’s write out the first 4 terms.

The pattern is clear (and as a challenge, prove to me in the comments that the pattern holds true for later terms):

The sequence goes like -3, 3, -3, 3, -3, 3, and so on.

So the sum of each pair of terms is zero.

So we can sum up the terms as follows: