Question

How do you use the binomial theorem to approximate the value of `1.07^7` ?

Answer 1

`(1.07)^7 ~~ 1.60578`

Explanation:

By the binomial theorem we have:

`(a+b)^7 = sum_(k=0)^7 ((7),(k)) a^(7-k)b^k`

where `((7),(k)) = (7!)/(k!(7-k)!)`

We can get these binomial coefficients from the row of Pascal's triangle that begins `1, 7`. Some people call this the `7`th row (calling the first row the `0`th). Personally I prefer to call it the `8`th row, but regardless of what you call it, it's the one that begins `1, 7`:

So:

`(a+b)^7 = a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7`

Putting `a=1` and `b=0.07` we have:

`1.07^7 ~~ 1+7(0.07)+21(0.07)^2+35(0.07)^3+35(0.07)^4+21(0.07)^5`

`=1+7(0.07)+21(0.0049)+35(0.000343)+35(0.00002401)+21(0.0000016807)`

`=1+0.49+0.1029+0.012005+0.00084035+0.0000352947`

`~~1.60578`