Solve 1/2012*(1-1/2013)*(1-1/2014)*(1-1/2015)*(1-frac{1}{2016})*(1-frac{1}{2017})

$$\frac { 1 } { 2012 } \times ( 1 - \frac { 1 } { 2013 } ) \times ( 1 - \frac { 1 } { 2014 } ) \times ( 1 - \frac { 1 } { 2015 } ) \times ( 1 - \frac { 1 } { 2016 } ) \times ( 1 - \frac { 1 } { 2017 } )$$

Evaluate

$$\frac{1}{2017}\approx 0.000495786$$

Solution Steps

Convert 1 to fraction \frac{2013}{2013}.

$$\frac{1}{2012}\left(\frac{2013}{2013}-\frac{1}{2013}\right)\left(1-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Since \frac{2013}{2013} and \frac{1}{2013} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2012}\times \frac{2013-1}{2013}\left(1-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Subtract 1 from 2013 to get 2012.

$$\frac{1}{2012}\times \frac{2012}{2013}\left(1-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Multiply \frac{1}{2012} times \frac{2012}{2013} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2012}{2012\times 2013}\left(1-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Cancel out 2012 in both numerator and denominator.

$$\frac{1}{2013}\left(1-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Convert 1 to fraction \frac{2014}{2014}.

$$\frac{1}{2013}\left(\frac{2014}{2014}-\frac{1}{2014}\right)\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Since \frac{2014}{2014} and \frac{1}{2014} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2013}\times \frac{2014-1}{2014}\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Subtract 1 from 2014 to get 2013.

$$\frac{1}{2013}\times \frac{2013}{2014}\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Multiply \frac{1}{2013} times \frac{2013}{2014} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2013}{2013\times 2014}\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Cancel out 2013 in both numerator and denominator.

$$\frac{1}{2014}\left(1-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Convert 1 to fraction \frac{2015}{2015}.

$$\frac{1}{2014}\left(\frac{2015}{2015}-\frac{1}{2015}\right)\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Since \frac{2015}{2015} and \frac{1}{2015} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2014}\times \frac{2015-1}{2015}\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Subtract 1 from 2015 to get 2014.

$$\frac{1}{2014}\times \frac{2014}{2015}\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Multiply \frac{1}{2014} times \frac{2014}{2015} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2014}{2014\times 2015}\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Cancel out 2014 in both numerator and denominator.

$$\frac{1}{2015}\left(1-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Convert 1 to fraction \frac{2016}{2016}.

$$\frac{1}{2015}\left(\frac{2016}{2016}-\frac{1}{2016}\right)\left(1-\frac{1}{2017}\right)$$

Since \frac{2016}{2016} and \frac{1}{2016} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2015}\times \frac{2016-1}{2016}\left(1-\frac{1}{2017}\right)$$

Subtract 1 from 2016 to get 2015.

$$\frac{1}{2015}\times \frac{2015}{2016}\left(1-\frac{1}{2017}\right)$$

Multiply \frac{1}{2015} times \frac{2015}{2016} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2015}{2015\times 2016}\left(1-\frac{1}{2017}\right)$$

Cancel out 2015 in both numerator and denominator.

$$\frac{1}{2016}\left(1-\frac{1}{2017}\right)$$

Convert 1 to fraction \frac{2017}{2017}.

$$\frac{1}{2016}\left(\frac{2017}{2017}-\frac{1}{2017}\right)$$

Since \frac{2017}{2017} and \frac{1}{2017} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2016}\times \frac{2017-1}{2017}$$

Subtract 1 from 2017 to get 2016.

$$\frac{1}{2016}\times \frac{2016}{2017}$$

Multiply \frac{1}{2016} times \frac{2016}{2017} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 2016}{2016\times 2017}$$

Cancel out 2016 in both numerator and denominator.

$$\frac{1}{2017}$$

Factor

$$\frac{1}{2017} = 0.0004957858205255329$$