Question

What are the values of the following products?

$$\begin{gathered} \mathrm{a})\mathbf{\prod }_{\mathbf{i}\mathbf{=}\mathbf{0}}^{\mathbf{10}}\mathbf{ }\mathbf{i} \\ \mathbf{b}\mathbf{)}\mathbf{}\mathbf{}\mathbf{\prod }_{\mathbf{i}\mathbf{=}\mathbf{5}}^{\mathbf{10}}\mathbf{ }\mathbf{i} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{}\mathbf{\prod }_{\mathbf{i}\mathbf{=}\mathbf{1}}^{\mathbf{100}}\mathbf{ }{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{i}}\mathbf{}\mathbf{d}\mathbf{)}\mathbf{}\mathbf{}\mathbf{\prod }_{\mathbf{i}\mathbf{=}\mathbf{1}}^{\mathbf{10}}\mathbf{ }\mathbf{2} \end{gathered}$$

Short answer

(a) The value of product is 0.

(b) The value of product is 1680 .

(c) The value of product is 1.

(d) The value of product is 1024.

Step-by-step solution

Determination of value of product(a)

The product sigma gives the multiplication of all the numbers from lower limit to upper limit.

The value of the product is given as:

$\begin{array}{r}\prod _{\mathrm{i}=0}^{10} \mathrm{i}=\left(0\right)\left(1\right)\left(2\right)\left(3\right)\left(4\right)\left(5\right)\left(6\right)\left(7\right)\left(8\right)\left(9\right)\left(10\right)\\ \prod _{\mathrm{i}=0}^{10} \mathrm{i}=0\end{array}$

Therefore, the value of product is 0 .

Determination of value of product(b)

The value of the product is given as:

$\begin{array}{r}\prod _{i=5}^8 i=\left(5\right)\left(6\right)\left(7\right)\left(8\right)\\ \prod _{i=5}^8 i=1680\end{array}$

Therefore, the value of product is 1680 .

Determination of value of product(c)

The value of the product is given as:

$\begin{array}{r}\prod _{i=1}^{100} (-1{)}^i=(-1{)}^1(-1{)}^2(-1{)}^3(-1{)}^4\dots \dots \dots ..(-1{)}^{97}(-1{)}^{98}(-1{)}^{99}(-1{)}^{100}\\ \prod _{i=1}^{100} (-1{)}^i=\left[(-1{)}^1(-1{)}^3(-1{)}^5\dots \dots (-1{)}^{97}(-1{)}^{99}\right]\left[(-1{)}^2(-1{)}^4(-1{)}^6\dots \dots .(-1{)}^{98}(-1{)}^{100}\right]\\ \prod _{i=1}^{100} (-1{)}^i=(1\left)\right(1)\\ \prod _{i=1}^{100} (-1{)}^i=1\end{array}$

Therefore, the value of product is 1 .

Determination of value of product(d)

The value of the product is given as:

$\begin{array}{r}\prod _{i=1}^{10} 2=\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\\ =2^{10}\\ =1024\end{array}$

Therefore, the value of product is 1024 .