Solve frac{3i^{2019}+4i^2018+5i^2017+6i^2016+7i^2015}{1+i}

$$\frac { 3 i ^ { 2019 } + 4 i ^ { 2018 } + 5 i ^ { 2017 } + 6 i ^ { 2016 } + 7 i ^ { 2015 } } { 1 + i }$$

Evaluate

$$-\frac{3}{2}-\frac{7}{2}i=-1.5-3.5i$$

Solution Steps

Calculate i to the power of 2019 and get -i.

$$\frac{3\left(-i\right)+4i^{2018}+5i^{2017}+6i^{2016}+7i^{2015}}{1+i}$$

Multiply 3 and -i to get -3i.

$$\frac{-3i+4i^{2018}+5i^{2017}+6i^{2016}+7i^{2015}}{1+i}$$

Calculate i to the power of 2018 and get -1.

$$\frac{-3i+4\left(-1\right)+5i^{2017}+6i^{2016}+7i^{2015}}{1+i}$$

Multiply 4 and -1 to get -4.

$$\frac{-3i-4+5i^{2017}+6i^{2016}+7i^{2015}}{1+i}$$

Calculate i to the power of 2017 and get i.

$$\frac{-3i-4+5i+6i^{2016}+7i^{2015}}{1+i}$$

Do the additions in -3i-4+5i.

$$\frac{6i^{2016}+7i^{2015}-4+2i}{1+i}$$

Calculate i to the power of 2016 and get 1.

$$\frac{6\times 1+7i^{2015}-4+2i}{1+i}$$

Multiply 6 and 1 to get 6.

$$\frac{6+7i^{2015}-4+2i}{1+i}$$

Do the additions in 6-4+2i.

$$\frac{7i^{2015}+2+2i}{1+i}$$

Calculate i to the power of 2015 and get -i.

$$\frac{7\left(-i\right)+2+2i}{1+i}$$

Multiply 7 and -i to get -7i.

$$\frac{-7i+2+2i}{1+i}$$

Do the additions in -7i+2+2i.

$$\frac{2-5i}{1+i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 1-i.

$$\frac{\left(2-5i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}$$

Do the multiplications in \frac{\left(2-5i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}.

$$\frac{-3-7i}{2}$$

Divide -3-7i by 2 to get -\frac{3}{2}-\frac{7}{2}i.

$$-\frac{3}{2}-\frac{7}{2}i$$

Real Part

$$-\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Solution Steps

Calculate i to the power of 2019 and get -i.

$$Re(\frac{3\left(-i\right)+4i^{2018}+5i^{2017}+6i^{2016}+7i^{2015}}{1+i})$$

Multiply 3 and -i to get -3i.

$$Re(\frac{-3i+4i^{2018}+5i^{2017}+6i^{2016}+7i^{2015}}{1+i})$$

Calculate i to the power of 2018 and get -1.

$$Re(\frac{-3i+4\left(-1\right)+5i^{2017}+6i^{2016}+7i^{2015}}{1+i})$$

Multiply 4 and -1 to get -4.

$$Re(\frac{-3i-4+5i^{2017}+6i^{2016}+7i^{2015}}{1+i})$$

Calculate i to the power of 2017 and get i.

$$Re(\frac{-3i-4+5i+6i^{2016}+7i^{2015}}{1+i})$$

Do the additions in -3i-4+5i.

$$Re(\frac{6i^{2016}+7i^{2015}-4+2i}{1+i})$$

Calculate i to the power of 2016 and get 1.

$$Re(\frac{6\times 1+7i^{2015}-4+2i}{1+i})$$

Multiply 6 and 1 to get 6.

$$Re(\frac{6+7i^{2015}-4+2i}{1+i})$$

Do the additions in 6-4+2i.

$$Re(\frac{7i^{2015}+2+2i}{1+i})$$

Calculate i to the power of 2015 and get -i.

$$Re(\frac{7\left(-i\right)+2+2i}{1+i})$$

Multiply 7 and -i to get -7i.

$$Re(\frac{-7i+2+2i}{1+i})$$

Do the additions in -7i+2+2i.

$$Re(\frac{2-5i}{1+i})$$

Multiply both numerator and denominator of \frac{2-5i}{1+i} by the complex conjugate of the denominator, 1-i.

$$Re(\frac{\left(2-5i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)})$$

Do the multiplications in \frac{\left(2-5i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}.

$$Re(\frac{-3-7i}{2})$$

Divide -3-7i by 2 to get -\frac{3}{2}-\frac{7}{2}i.

$$Re(-\frac{3}{2}-\frac{7}{2}i)$$

The real part of -\frac{3}{2}-\frac{7}{2}i is -\frac{3}{2}.

$$-\frac{3}{2}$$