Binomial expansion for approximation
I've just had to do a homework on binomial expansion for approximation:
$1.07^9$
so:
$(1+0.07)^9$
To do binomial expansion you need a calculator for the combinations button (nCr), so why would use a more complicated method, which only gives an approximation be used over just typing 1.07^9 into a calculator?
(or is this never done in real life, and it's just a homework?)
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$\begingroup$Expanding the whole thing using Binomial Theorem gives you an exact value. Not an approximation.
To get an approximation you can consider a few terms from the expansion.
For instance, for "small" $x$, $1+nx$ is a "reasonable" approximation for $(1+x)^n$.
Notice that this corresponds to picking the first two terms from the binomial theorem expansion $(1+x)^n = 1 + \binom{n}{1} \ x + \binom{n}{2}\ x^2 + \dots + x^n$.
For example
$1.0007^9 \approx 1 + 9\times 0.0007 = 1.0063$ which agrees with $1.0007^9 = 1.0063176688422737867054812736724$ upto $4$ decimal places.
Depending on how accurate you want it, you could consider more terms from the binomial expansion.
This is based on the fact that for small $x$, as the power $r$ of $x$ gets larger, the term $x^r$ becomes small quite fast.
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