Question

Explain how you can use the terms from the binomial expansion to approximate 4.025. 1st term: 1024 2nd term: 25.6 3rd term: 0.256 4th term: 0.001

Answer 1

Answer:

  add up the given terms: 1024 + 25.6 + 0.256 + 0.001 = 1049.857

Step-by-step explanation:

The 5th power of a binomial is ...

$(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5$

For a=4 and b=0.02, the expansion is ...

$4.02^5=4^5+5\cdot 4^4\cdot 0.02+10\cdot 4^3\cdot 0.02^2+10\cdot 4^2\cdot 0.02^3+5\cdot 4\cdot 0.02^4 +0.02^5\\\\=1024+25.6+0.256+0.00128+0.0000032+0.0000000032$

Rounding to thousandths, the series is effectively truncated at 4 terms, so the 5th power of 4.02 is about ...

  1024 +25.6 +0.256 +0.001 = 1049.857

Answer 2

The 4th term is almost 0, so the 5th and 6th terms are also close to 0. Add the terms shown to approximate 4.025:  1024 + 25.6 + 0.256 + 0.001 = 1049.857