Binomial Estimation
Find the first 5 items in ascending powers of x of \(\left( 3+\frac{x}{4} \right)^{11}\). Hence find an estimate for the value of \(3.002^{11}\)
- \((3+\frac{x}{4})^{11} = 177147+162384.75x+67660.3125x^2+16915.07813x^3+2819.179688x^4\)
- If \(\frac{x}{4}=0.002\) then \(x=0.008\)
- \(177147+(162384.75\times0.008)+\dots\)
- \(177147+1299.078+4.33026\dots\)
- \(178450.4169\)
Problems (Ex 8E)
Find the first 5 terms of the binomial expansion for \(\left( 1-\frac{x}{10} \right)^6\)
- \(1^6+\left( ^6C_{1} \times 1^5\times \frac{x}{10} \right)+\left( ^6C_2\times{1}^4\times \frac{x}{10}^2 \right)+\left( ^6C_{3}\times{1}^3\times \frac{x}{10}^3 \right)\)
- \(1+\left( 6\times{1}\times \frac{x}{10} \right)+\left( 15\times 1\times \frac{x}{100} \right)+\left( 20\times{1}\times \frac{x}{1000} \right)\)
- \(1+0.6x+0.15x^2+0.02x^3\)
Hence find an estimate for \(0.99^6\) - \(x={0.1}\)
- \(0.9 41 48\)
Find the first 4 powers of x for the binomial expansion of \(\left( 2+\frac{x}{5} \right)^{10}\)
- \(2^{10}+\left( ^{10}C_{1}\times{2}^9\times \frac{x}{5} \right)+\left( ^{10}C_{2}\times{2}^8\times \frac{x}{5}^2 \right)+\left( ^{10}C_{3}\times 2^8\times \frac{x}{5}^3 \right)\)
- \(1024+\)
UNFINISHED