Use Newton\'s binomial theorem to approximate sqrt 30 Solution (1+Q)^(m/n) = 1 +
ID: 2943017 • Letter: U
Question
Use Newton's binomial theorem to approximate sqrt 30Explanation / Answer
(1+Q)^(m/n) = 1 +(m/n)Q + [(m/n)*(m/n - 1)/2!]*Q^2 +[(m/n)*(m/n-1)*(m/n - 2)/3!]*Q^3 = ... m=1, n=2 In our problem Q = 29, this formula blows up if Q >1 What do we do? The largest perfect square under thirty is 25, we can takes this out. SQRT(30) = SQRT25*SQRT(30/25) = SQRT25*SQRT(6/5) = SQRT(25)*SQRT(5/5+1/5) So now we will multiply our answer by 5 and solve (1 + 1/5)^1/2 Here are the first ten terms. The last column is the sum of the terms so far. We can see that it converges to 1.0954451 (this is 1/5 of the answer) 1 1 1 1 2 0.5 0.1 1.1 3 -0.5 -0.005 1.095 4 -1.5 0.0005 1.0955 5 -2.5 -0.0000625 1.0954375 6 -3.5 0.00000875 1.09544625 7 -4.5 -1.3125E-06 1.095444938 8 -5.5 2.0625E-07 1.095445144 9 -6.5 -3.35156E-08 1.09544511 10 -7.5 5.58594E-09 1.095445116 Now we multiply by SQRT(25) or 5 5*1.095445115 = 5.477225575 When I solve for SQRT(30) in Excel, I get 5.477225575, so the limit of this apporximation (36 terms) is the same as the value out to arbitrary number of digits.
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