Solve (1+1/2)(2+3/4)+3+5/6)dots(n+2n-1/2n)=

$$( 1 + \frac { 1 } { 2 } ) ( 2 + \frac { 3 } { 4 } ) + 3 + \frac { 5 } { 6 } ) \dots ( n + \frac { 2 n - 1 } { 2 n } ) =$$

Evaluate

$$\frac{191}{24}\approx 7.958333333$$

Solution Steps

Convert 1 to fraction \frac{2}{2}.

$$\left(\frac{2}{2}+\frac{1}{2}\right)\left(2+\frac{3}{4}\right)+3+\frac{5}{6}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{2}\left(2+\frac{3}{4}\right)+3+\frac{5}{6}$$

Add 2 and 1 to get 3.

$$\frac{3}{2}\left(2+\frac{3}{4}\right)+3+\frac{5}{6}$$

Convert 2 to fraction \frac{8}{4}.

$$\frac{3}{2}\left(\frac{8}{4}+\frac{3}{4}\right)+3+\frac{5}{6}$$

Since \frac{8}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\frac{3}{2}\times \frac{8+3}{4}+3+\frac{5}{6}$$

Add 8 and 3 to get 11.

$$\frac{3}{2}\times \frac{11}{4}+3+\frac{5}{6}$$

Multiply \frac{3}{2} times \frac{11}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\times 11}{2\times 4}+3+\frac{5}{6}$$

Do the multiplications in the fraction \frac{3\times 11}{2\times 4}.

$$\frac{33}{8}+3+\frac{5}{6}$$

Convert 3 to fraction \frac{24}{8}.

$$\frac{33}{8}+\frac{24}{8}+\frac{5}{6}$$

Since \frac{33}{8} and \frac{24}{8} have the same denominator, add them by adding their numerators.

$$\frac{33+24}{8}+\frac{5}{6}$$

Add 33 and 24 to get 57.

$$\frac{57}{8}+\frac{5}{6}$$

Least common multiple of 8 and 6 is 24. Convert \frac{57}{8} and \frac{5}{6} to fractions with denominator 24.

$$\frac{171}{24}+\frac{20}{24}$$

Since \frac{171}{24} and \frac{20}{24} have the same denominator, add them by adding their numerators.

$$\frac{171+20}{24}$$

Add 171 and 20 to get 191.

$$\frac{191}{24}$$

Factor

$$\frac{191}{2 ^ {3} \cdot 3} = 7\frac{23}{24} = 7.958333333333333$$