Solve 1/3+1/3^{2}+1/3^{3}+1/3^4+frac{1}{3^5}+frac{1}{3^6}+frac{1}{3^7}=

$$\frac { 1 } { 3 } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 3 ^ { 4 } } + \frac { 1 } { 3 ^ { 5 } } + \frac { 1 } { 3 ^ { 6 } } + \frac { 1 } { 3 ^ { 7 } } =$$

Evaluate

$$\frac{1093}{2187}\approx 0.499771376$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{3}+\frac{1}{9}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Least common multiple of 3 and 9 is 9. Convert \frac{1}{3} and \frac{1}{9} to fractions with denominator 9.

$$\frac{3}{9}+\frac{1}{9}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Since \frac{3}{9} and \frac{1}{9} have the same denominator, add them by adding their numerators.

$$\frac{3+1}{9}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Add 3 and 1 to get 4.

$$\frac{4}{9}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4}{9}+\frac{1}{27}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Least common multiple of 9 and 27 is 27. Convert \frac{4}{9} and \frac{1}{27} to fractions with denominator 27.

$$\frac{12}{27}+\frac{1}{27}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Since \frac{12}{27} and \frac{1}{27} have the same denominator, add them by adding their numerators.

$$\frac{12+1}{27}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Add 12 and 1 to get 13.

$$\frac{13}{27}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{13}{27}+\frac{1}{81}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Least common multiple of 27 and 81 is 81. Convert \frac{13}{27} and \frac{1}{81} to fractions with denominator 81.

$$\frac{39}{81}+\frac{1}{81}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Since \frac{39}{81} and \frac{1}{81} have the same denominator, add them by adding their numerators.

$$\frac{39+1}{81}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Add 39 and 1 to get 40.

$$\frac{40}{81}+\frac{1}{3^{5}}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{40}{81}+\frac{1}{243}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Least common multiple of 81 and 243 is 243. Convert \frac{40}{81} and \frac{1}{243} to fractions with denominator 243.

$$\frac{120}{243}+\frac{1}{243}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Since \frac{120}{243} and \frac{1}{243} have the same denominator, add them by adding their numerators.

$$\frac{120+1}{243}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Add 120 and 1 to get 121.

$$\frac{121}{243}+\frac{1}{3^{6}}+\frac{1}{3^{7}}$$

Calculate 3 to the power of 6 and get 729.

$$\frac{121}{243}+\frac{1}{729}+\frac{1}{3^{7}}$$

Least common multiple of 243 and 729 is 729. Convert \frac{121}{243} and \frac{1}{729} to fractions with denominator 729.

$$\frac{363}{729}+\frac{1}{729}+\frac{1}{3^{7}}$$

Since \frac{363}{729} and \frac{1}{729} have the same denominator, add them by adding their numerators.

$$\frac{363+1}{729}+\frac{1}{3^{7}}$$

Add 363 and 1 to get 364.

$$\frac{364}{729}+\frac{1}{3^{7}}$$

Calculate 3 to the power of 7 and get 2187.

$$\frac{364}{729}+\frac{1}{2187}$$

Least common multiple of 729 and 2187 is 2187. Convert \frac{364}{729} and \frac{1}{2187} to fractions with denominator 2187.

$$\frac{1092}{2187}+\frac{1}{2187}$$

Since \frac{1092}{2187} and \frac{1}{2187} have the same denominator, add them by adding their numerators.

$$\frac{1092+1}{2187}$$

Add 1092 and 1 to get 1093.

$$\frac{1093}{2187}$$

Factor

$$\frac{1093}{3 ^ {7}} = 0.4997713763145862$$