Solve 0.11/1.0552+0.11/1.0552^2+0.11/1.0552^3+0.11left(1+0.0286right)/1.0552^3left(0.0552-0.0286right)

$$\frac{ 0.11 }{ 1.0552 } + \frac{ 0.11 }{ { 1.0552 }^{ 2 } } + \frac{ 0.11 }{ { 1.0552 }^{ 3 } } + \frac{ 0.11 \left( 1+0.0286 \right) }{ { 1.0552 }^{ 3 } \left( 0.0552-0.0286 \right) }$$

Evaluate

$$\frac{1812711175}{462776426}\approx 3.917034389$$

Solution Steps

Expand \frac{0.11}{1.0552} by multiplying both numerator and the denominator by 10000.

$$\frac{1100}{10552}+\frac{0.11}{1.0552^{2}}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Reduce the fraction \frac{1100}{10552} to lowest terms by extracting and canceling out 4.

$$\frac{275}{2638}+\frac{0.11}{1.0552^{2}}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Calculate 1.0552 to the power of 2 and get 1.11344704.

$$\frac{275}{2638}+\frac{0.11}{1.11344704}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Expand \frac{0.11}{1.11344704} by multiplying both numerator and the denominator by 100000000.

$$\frac{275}{2638}+\frac{11000000}{111344704}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Reduce the fraction \frac{11000000}{111344704} to lowest terms by extracting and canceling out 64.

$$\frac{275}{2638}+\frac{171875}{1739761}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Least common multiple of 2638 and 1739761 is 3479522. Convert \frac{275}{2638} and \frac{171875}{1739761} to fractions with denominator 3479522.

$$\frac{362725}{3479522}+\frac{343750}{3479522}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Since \frac{362725}{3479522} and \frac{343750}{3479522} have the same denominator, add them by adding their numerators.

$$\frac{362725+343750}{3479522}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Add 362725 and 343750 to get 706475.

$$\frac{706475}{3479522}+\frac{0.11}{1.0552^{3}}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Calculate 1.0552 to the power of 3 and get 1.174909316608.

$$\frac{706475}{3479522}+\frac{0.11}{1.174909316608}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Expand \frac{0.11}{1.174909316608} by multiplying both numerator and the denominator by 1000000000000.

$$\frac{706475}{3479522}+\frac{110000000000}{1174909316608}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Reduce the fraction \frac{110000000000}{1174909316608} to lowest terms by extracting and canceling out 512.

$$\frac{706475}{3479522}+\frac{214843750}{2294744759}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Least common multiple of 3479522 and 2294744759 is 4589489518. Convert \frac{706475}{3479522} and \frac{214843750}{2294744759} to fractions with denominator 4589489518.

$$\frac{931840525}{4589489518}+\frac{429687500}{4589489518}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Since \frac{931840525}{4589489518} and \frac{429687500}{4589489518} have the same denominator, add them by adding their numerators.

$$\frac{931840525+429687500}{4589489518}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Add 931840525 and 429687500 to get 1361528025.

$$\frac{1361528025}{4589489518}+\frac{0.11\left(1+0.0286\right)}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Add 1 and 0.0286 to get 1.0286.

$$\frac{1361528025}{4589489518}+\frac{0.11\times 1.0286}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Multiply 0.11 and 1.0286 to get 0.113146.

$$\frac{1361528025}{4589489518}+\frac{0.113146}{1.0552^{3}\left(0.0552-0.0286\right)}$$

Calculate 1.0552 to the power of 3 and get 1.174909316608.

$$\frac{1361528025}{4589489518}+\frac{0.113146}{1.174909316608\left(0.0552-0.0286\right)}$$

Subtract 0.0286 from 0.0552 to get 0.0266.

$$\frac{1361528025}{4589489518}+\frac{0.113146}{1.174909316608\times 0.0266}$$

Multiply 1.174909316608 and 0.0266 to get 0.0312525878217728.

$$\frac{1361528025}{4589489518}+\frac{0.113146}{0.0312525878217728}$$

Expand \frac{0.113146}{0.0312525878217728} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{1361528025}{4589489518}+\frac{1131460000000000}{312525878217728}$$

Reduce the fraction \frac{1131460000000000}{312525878217728} to lowest terms by extracting and canceling out 1024.

$$\frac{1361528025}{4589489518}+\frac{1104941406250}{305201052947}$$

Least common multiple of 4589489518 and 305201052947 is 610402105894. Convert \frac{1361528025}{4589489518} and \frac{1104941406250}{305201052947} to fractions with denominator 610402105894.

$$\frac{181083227325}{610402105894}+\frac{2209882812500}{610402105894}$$

Since \frac{181083227325}{610402105894} and \frac{2209882812500}{610402105894} have the same denominator, add them by adding their numerators.

$$\frac{181083227325+2209882812500}{610402105894}$$

Add 181083227325 and 2209882812500 to get 2390966039825.

$$\frac{2390966039825}{610402105894}$$

Reduce the fraction \frac{2390966039825}{610402105894} to lowest terms by extracting and canceling out 1319.

$$\frac{1812711175}{462776426}$$

Factor

$$\frac{11 \cdot 6591677 \cdot 5 ^ {2}}{2 \cdot 7 \cdot 19 \cdot 1319 ^ {2}} = 3\frac{424381897}{462776426} = 3.917034388869238$$