## Compound Interest and Pascal’s Triangle

Today I was observing an NQT and the lesson was on compound interest. His question to introduce the subject was “if you invested £1000 at 10% apr for 3 years how much would you have at the end?” I thought it was a great way to introduce it and it’s similar to what I do, although I must never have used that combination of numbers, because I noticed something rather interesting that I’d never noticed before.

After some thinking time, a number of pupils declaring that £1300 was the answer and a discussion around how this sort of interest is actually calculated, he talked through the answer on the board:

*Start £1000*

*1 year £1100*

*2 years £1210*

*3 years £1331*

Have you noticed what I did?

At this point in the lesson I thought, it’s **Pascal’s Triangle! **I quickly checked the next number and of course I came up with £1464.10. I then realised that the place value of our decimal system would mean it would go a bit skewiff here after.

I thought about why this was. Before I thought this was Pascal’s Triangle I had thought, “they’re the binomial coefficients!” So I figured this must have something to do with it. I thought about the sum:

1000 x 1.1^n when n is number of years

I realised that could be written

1000 x (1 + 0.1)^n

This means that each coefficient is multiplied by a different power of ten, and thus explains why we see the coefficients. But more importantly it means:

*Any compound interest problem can be reduced to a binomial expansion problem!*

c x (1 + i)^n where c is capital, i is interest (as a decimal) and n the number of periods.

I know that this will be known by some already, but I had never made this link before and found it fascinating, and hope you do too. It’s just another example of the beauty and connectivity of maths.

I’d love to hear if you have ever used this link before, or if you yourself have noticed anything cool like this.

Can’t say I’ve noticed that but it’s pretty cool.

Good spot.

Cheers, I thought so too!

Excellent spot. I love teaching compound interest. A good starter I find to draw out the misconception is “Dave invested £1000 with an annual interest rate of 10%. Dave says he will double his money in ten years. Do you agree”!

Aye, that’s a great ne to start with!