Need urgent help understanding binomial expansion

Ive been thrown into my first calc class at college and I cant understand how to do this: "Use the binomial expansion formula to find (100.1)^4, accurate to the ten-thousands."

  • Erdos Erdos

    What do you mean by "(100.1)4"? Is there a typo?

    • I meant to put (100.1)^4 as 4 as the exponent. That is how the question is exactly phrased on the homework and its hurting my head

  • Questions at this level should come with a bounty.

    • Is it that hard compared to other questions on here? Cause this looks like the least complex compared to others. I can add one but didnt know how it worked

    • Answering this question carefully would take about 20 minutes. You should offer a bounty so one has the incentive to answer. Also many users do not get notifications for Pro Bono questions, and hence they are less likely to get answered. You should repost the question if you want to offer a bounty.

1 Answer

By binomial expansion formula, for any integer $n$,  we have 
\[(a+x)^{n}=\sum_{k=0}^{n}{n\choose k}x^{k}a^{n-k},\]
If we only use $m-1$ terms for the approximation, then the error will be bounded by 
\[\text{Error} \leq {n\choose m}x^{m}c^{n-m},\]
for some $0\leq c \leq x$.

In this example $a=100$, $n=4$, and $x=0.01$. So  for the approximation to be accurate to the ten-thousands we need to have 

\[\text{Error} \leq {4\choose m}x^{m}c^{4-m}\leq {4\choose m}(0.01)^{m}(100)^{4-m} <\frac{1}{10000}.\]

You can check that the smallest $m$ for which the above ineauality holds is $m=4$. So we should use $m-1=3$ terms in the binomial expansion to find $(100.1)^4$, accurate to the ten-thousands. Hence
\[(100+.01)^{n}\approx\sum_{k=0}^{3}{n\choose k}x^{k}a^{n-k}\]
\[={4\choose 0} (0.01)^{0}(100)^{4-0}+{4\choose 1} (0.01)^{1}(100)^{4-1}+{4\choose 2} (0.01)^{2}(100)^{4-2}+{4\choose 3} (0.01)^{3}(100)^{4-3} \]
\[=(100)^4+4(100)^2+6+4\times (0.01)^2\]

  • This took me over 35 minutes to answer. I was hesitant to answer, but I thought this might be the first time you are asking a questions here, so I decided to answer. I would appreciate a fair tip! Your should offer a fair bounty for your future questions, otherwise you may not get an answer.

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