Solve D(3)=(1/9+1)^2*1/3+(1/9)*1/3+(frac{1}{9}-1)^2*frac{2}{9}+(frac{1}{9}-2)*frac{1}{9}

$$D ( 3 ) = ( \frac { 1 } { 9 } + 1 ) ^ { 2 } \times \frac { 1 } { 3 } + ( \frac { 1 } { 9 } ) \times \frac { 1 } { 3 } + ( \frac { 1 } { 9 } - 1 ) ^ { 2 } \times \frac { 2 } { 9 } + ( \frac { 1 } { 9 } - 2 ) \times \frac { 1 } { 9 }$$

Evaluate

$$\frac{302}{729}\approx 0.414266118$$

Solution Steps

Add \frac{1}{9} and 1 to get \frac{10}{9}.

$$\left(\frac{10}{9}\right)^{2}\times \frac{1}{3}+\frac{1}{9}\times \frac{1}{3}+\left(\frac{1}{9}-1\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Calculate \frac{10}{9} to the power of 2 and get \frac{100}{81}.

$$\frac{100}{81}\times \frac{1}{3}+\frac{1}{9}\times \frac{1}{3}+\left(\frac{1}{9}-1\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Multiply \frac{100}{81} and \frac{1}{3} to get \frac{100}{243}.

$$\frac{100}{243}+\frac{1}{9}\times \frac{1}{3}+\left(\frac{1}{9}-1\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Multiply \frac{1}{9} and \frac{1}{3} to get \frac{1}{27}.

$$\frac{100}{243}+\frac{1}{27}+\left(\frac{1}{9}-1\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Add \frac{100}{243} and \frac{1}{27} to get \frac{109}{243}.

$$\frac{109}{243}+\left(\frac{1}{9}-1\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Subtract 1 from \frac{1}{9} to get -\frac{8}{9}.

$$\frac{109}{243}+\left(-\frac{8}{9}\right)^{2}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Calculate -\frac{8}{9} to the power of 2 and get \frac{64}{81}.

$$\frac{109}{243}+\frac{64}{81}\times \frac{2}{9}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Multiply \frac{64}{81} and \frac{2}{9} to get \frac{128}{729}.

$$\frac{109}{243}+\frac{128}{729}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Add \frac{109}{243} and \frac{128}{729} to get \frac{455}{729}.

$$\frac{455}{729}+\left(\frac{1}{9}-2\right)\times \frac{1}{9}$$

Subtract 2 from \frac{1}{9} to get -\frac{17}{9}.

$$\frac{455}{729}-\frac{17}{9}\times \frac{1}{9}$$

Multiply -\frac{17}{9} and \frac{1}{9} to get -\frac{17}{81}.

$$\frac{455}{729}-\frac{17}{81}$$

Subtract \frac{17}{81} from \frac{455}{729} to get \frac{302}{729}.

$$\frac{302}{729}$$

Factor

$$\frac{2 \cdot 151}{3 ^ {6}} = 0.41426611796982166$$