Question

How do you use the binomial theorem to calculate `1.01^5` ?

Answer 1

`1.01^5 = 1.0510100501`

Explanation:

By the binomial theorem, in general we have:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k`

where `((n),(k)) = (n!)/((n-k)! k!)`

The binomial coefficient `((n), (k))` occurs as the `(k+1)`th term in the `(n+1)`th row of Pascal's triangle. in our example, we are interested in the sixth row, which consists of the terms:

`1, 5, 10, 10, 5, 1`

So we find:

`1.01^5 = (1+0.01)^5`

`color(white)(1.01^5) = 1+5(0.01)+10(0.01)^2+10(0.01)^3+5(0.01)^4+(0.01)^5`

`color(white)(1.01^5) = 1.0510100501`

Footnote

Note that for small powers `n`, this gives you a way to find a row of binomial coefficients on a calculator.

Just calculate `1.01^n` and read the digits in pairs. For example:

`1.01^4 = color(red)(1).color(green)(04)color(blue)(06)color(purple)(04)color(brown)(01)`