# Thinking before jumping

18 October 2012
Give me six hours to chop down a tree and I will spend the first four sharpening the axe.
Abraham Lincoln

I had a lesson which made me think of this quotation today. The class were looking at an interesting sequences problem which is stated below:
I’m walking down a long path in the park and I notice that the gardener has planted the trees and flowers next to the path in a particular order. He started with a tree, then planted two flowers, then a tree and then one flower, with this sequence continuing all the way along the path.

The path is 845 trees long. How many flowers are there along the path?

When some of the students in the room saw this problem, they of course said it was easy:
“All you have to do is 2 + 1 + 2 + 1 + 2 + 1 + … and keep on adding alternate two’s and one’s 844 times over.”

At this point at least 10 other students agreed and they started racing to see who could do the calculation first. I was nervous that no one would actually think a bit and use their knowledge of sequences to find a more efficient strategy.

Luckily for me one student took a different route. She drew down 5 trees with 6 flowers in between and then 10 trees with 14 flowers in between and then 15 trees with 21 flowers in between etc. until she had the following sequence:
Number of Trees:         5        10        15       20        25    …
Number of Plants:       6         14        21       29        36     …
+8        +7         +8         +7

At this point she paused. She put her hand up and said that she was frustrated because she thought she’d have a linear sequence which added the same amount every time. Funnily enough, as soon as she said the words out loud, a light bulb went off!

“I know! If you start with 5 trees and keep adding 10 trees each time then you will always add 15 plants – it actually is a linear sequence!”

Number of Trees:     5        15        25       35        45    …          Sequence 1
Number of Plants:   6         21        36       51        66     …         Sequence 2
+15      +15      +15       +15
Now I have this, I can find out which position the 845th tree is in Sequence 1 by doing:
10n – 10 = 845

n = 85th term

So the 845th tree is the 85th term in Sequence 1. Now I just have to find the 85th term of Sequence 2.

Sequence 2 has an nth term formula of 15n-9 and inserting n=85 gives the answer at 1266 plants.

Thankfully she saved my lesson. I thought for a minute that everyone would follow suit and calculate the number of flowers using the first approach. Anyhow, I think the other members of the class were about 10% through calculating when she said that she had an answer so we stopped there and I let her take it away. Needless to say, the rest of the class were impressed and I think they learnt something about sharpening the axe before starting to chop down the tree.

Dan Pearcy
(danpearcymaths.wordpress.com)