Sigma and Pi Notation for JEE

In mathematics, we use the capital “sigma” and “pi” notation to add and multiply elements of a sequence respectively. The Greek letter capital “sigma” or $\Sigma$ and capital “pi” or $\Pi$ are used with lower and upper limits of summation or multiplication. Moreover, lowercase “sigma” or $\sigma$ is used to denote “sigma-function” and as “standard deviation” in statistics. Lowercase “Pi” or $\pi$ is a universal constant with a value close to 3.14, used to measure the volume and circumference of cyclic objects. In mathematics, the notations Sigma (summation) and Pi (product) are used to express repeated addition or multiplication. When working with arithmetic or geometric series, the Sigma notation provides a succinct approach to describe multiple sums. Pi notation is a convenient technique to represent a wide range of products.

Sigma Symbol

An integer below the Sigma (the "beginning term number") and an integer above the Sigma (the "ending term number") are the common uses of Sigma notation. There are several sigma notations in mathematics. The popular ones are discussed below.

Summation Symbol

Suppose we have a sequence of numbers x1,x2,.......,xn and we want to add this sequence. We can write the sum as: 

• $x_{1}+x_{2}+\ldots \ldots+x_{n}=\sum_{i=1}^{n} x_{i}$

Example:

• $x_i=i^2$ and n=3

• $\sum_{i=1}^{3} i^{2}=1^{2}+2^{2}+3^{2}=1+4+9=14$

Some Properties of Summation

Here are some of the properties of summation notation:

• $\sum_{i=1}^{n} k x_{i}=k \sum_{i=1}^{n} x_{i}$ , where k is a constant.

Proof:

Given: the sequence xi and constant k

$\begin{align} &\sum_{i=1}^{n} k x_{i}=k x_{1}+k x_{2}+\ldots+k x_{n}=k\left(x_{1}+x_{2}+\ldots+x_{n}\right) \\ &\Rightarrow \sum_{i=1}^{n} k x_{i}=k \sum_{i=1}^{n} x_{i} \end{align}$

Hence proved.

• If xi=k, where k is a constant, then $\sum_{i=1}^{n} x_{i}=n \times x_{i}=n k$

Proof:

Given: the sequence xi,

$\begin{align} &x_{i}=k \\ &\sum_{i=1}^{n} x_{i}=\sum_{i=1}^{n} k=k+k+\ldots n \text { times }+k=n k \end{align}$

Hence proved.

• Suppose there are two sequences xi and yi then:

$\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} y_{i}$

Proof:

Given: two sequences xi and yi we get:

$\begin{align}&\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\left(x_{1}+y_{1}\right)+\left(x_{2}+y_{2}\right)+\ldots+\left(x_{n}+y_{n}\right)\\&\Rightarrow\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\left(x_{1}+x_{2}+\ldots+x_{n}\right)+\left(y_{1}+y_{2}+\ldots+y_{n}\right) \\ &\Rightarrow\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\sum_{i=1}^{n}\left(x_{i}\right)+\sum_{i=1}^{n}\left(y_{i}\right) \end{align}$

Hence proved.

• $\sum_{i=1}^{n}\left(x_{i}+a\right)=\sum_{i=1}^{n} x_{i}+n a$, where a is a constant.

Proof:

Using the above formula and with yi=a:

$\begin{align} &\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)=\sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} y_{i} \\ &\Rightarrow \sum_{i=1}^{n}\left(x_{i}+a\right)=\sum_{i=1}^{n} x_{i}+n a \end{align}$ Proved.

(applying the second property)

Sigma Symbol in Statistics

Lowercase “sigma” or $\sigma$ is used in statistics to denote standard deviation. We give you the formula for $\sigma$ below:

• $\sigma=\sqrt{\frac{\sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2}}{N}}$ where, $\mu=$mean and xi is the point in the sample space of N elements.

Sigma Function

The sigma function of a positive integer $n\in \mathbb{N}$ is the sum of the positive divisor of n. This is $\sigma(n)$ using the greek letter sigma. We can also calculate the sigma function for several values of n.

• $\begin{align} &\sigma(1)=1 \\ &\sigma(2)=3 \\ &\sigma(3)=4 \\ &\sigma(4)=7 \end{align}$ 

• For n=prime, $\sigma(n)=n+1$

• If $n=p_{1}^{k_{1}} p_{2}^{k_{2}} \ldots p_{3}^{k_{m}}$, where Pi=prime (This is basically prime decomposition of n) then, $\sigma(n)=\sigma\left(p_{1}^{k 1}\right) \sigma\left(p_{2}^{k_{2}}\right) \ldots \sigma\left(p_{m}^{k_{m}}\right)$

• Theorem: If p is prime and n is any positive integer, then $\sigma(p^n)$ is $\dfrac{\left(P^{n+1}-1\right)}{(p-1)}$.

Proof:

For a number pn, where p is a prime number, the divisors are: 1, p, p2, p3,.....pn

On addition we get:

$1+p+p^{2}+p^{3}+\ldots+p^{n}=\dfrac{p^{n+1}-1}{p-1}$

So, $\sigma(p)=\dfrac{p^{n+1}-1}{p-1}$

Hence proved.

We can use this theorem to calculate the sigma function of any $n\in \mathbb{N}$.

Pi Symbol

It is a mathematical symbol for the product of a group of terms. Suppose we have a sequence of numbers x1,x2,....xn and we want to multiply this sequence, then multiplication can be written as: 

$x_{1} \times x_{2} \times \ldots . \ldots \times x_{n}=\prod_{i=1}^{n} x_{i}$

Example:

xi=i and n=3

Then:

$\prod_{i=1}^{3} i=1 \times 2 \times 3=6$

Properties of Pi Symbol

Here we define a few properties of pi notation.

• Separation of Constant

$\prod_{i=1}^{n} c x_{i}=c^{n} \prod_{i=1}^{n} x_{i}$ , where c is a constant.

Proof:

Given:

$\begin{align} &\prod_{i=1}^{n} c x_{i}=\left(c x_{1}\right)\left(c x_{2}\right) \ldots n \text { times } .\left(c x_{n}\right) \\ &\Rightarrow \prod_{i=1}^{n} c x_{i}=c^{n}\left(x_{1} x_{2} \ldots x_{n}\right)=c^{n} \prod_{i=1}^{n} x_{i} \end{align}$

Hence proved.

• If xi and yi are two sequences of numbers, then:

$\prod_{i=1}^{n}\left(x_{i} y_{i}\right)=\left(\prod_{i=1}^{n} x_{i}\right)\left(\prod_{i=1}^{n} y_{i}\right)$

Proof:

Given:

$\prod_{i=1}^{n}\left(x_{i} y_{i}\right)=\left(x_{1} y_{1}\right)\left(x_{2} y_{2}\right) \ldots n \text { times } . .\left(x_{n} y_{n}\right)$

$\begin{align} &\Rightarrow \prod_{i=1}^{n}\left(x_{i} y_{i}\right)=\left(\left(x_{1}\right)\left(x_{2}\right) \ldots n \text { times.. }\left(x_{n}\right)\right)\left(\left(y_{1}\right)\left(y_{2}\right) \ldots n \text { times.. }\left(y_{n}\right)\right) \\&\Rightarrow \prod_{i=1}^{n}\left(x_{i} y_{i}\right)=\prod_{i=1}^{n}\left(x_{i}\right)+\prod_{i=1}^{n}\left(y_{i}\right) \end{align}$

Hence proved.

Quick Conversion of Summation to Pi Notation and Vice-versa

We will use log to convert summation to multiplication.

We know that:

$\log (A)+\log (B)=\log (A B)$

So:

$\log\left(x_{1}\right)+\log\left(x_{2}\right)+\ldots+\log\left(x_{n}\right)=\sum_{i=1}^{n} \log \left(x_{i}\right)=\log \left(\prod_{i=1}^{n} x_{i}\right)$

Now for the vice-versa case:

We know that:

$e^a\times e^b=e^{a+b}$

So:

$e^{x} \times e^{x^{2}} \times \ldots \times e^{x_{n}}=\prod_{i=1}^{n} e^{x_{i}}=e^{\sum_{i=1}^{n} x_{i}}$

Solved Example

1. We have one sequence of numbers {xi} and define another one by $a_{i}=x_{i+1}-x_{i}$ . Calculate$ \sum_{i=1}^{n} a_{i}$

Solution: Given: $a_{i}=x_{i+1}-x_{i}$

So: 

$\begin{align} &a_{1}=x_{2}-x_{1} \\ &a_{2}=x_{3}-x_{2} \\ &a_{3}=x_{4}-x_{3} \\.\\.\\.\\ &a_{n}=x_{n+1}-x_{n} \\ &\text { now we add it, } \\ &a_{1}+a_{2}+a_{3}+\ldots+a_{n}=\left(x_{2}-x_{1}\right)+\left(x_{3}-x_{2}\right)+\ldots+\left(x_{n+1}-x_{n}\right) \\ &\Rightarrow \sum_{i=1}^{n} a_{i}=x_{n+1}-x_{1} \end{align}$

This kind of sum is called Telescopic sum.

2. Find the sum of 1 + 4 + 7 + ... + (3n+1)

Solution. We can clearly see that each term of the sequence can be represented as:

$x_{i}=3 i+1$

So, the sum becomes:

• $\begin{align} &1+4+7+\ldots+(3 n+1)=\sum_{i=1}^{n}(3 i+1)=\sum_{i=1}^{n} 3 i+\sum_{i=1}^{n} 1 \\ &\Rightarrow 1+4+7+\ldots+(3 n+1)=3 \sum_{i=1}^{n} i+n \times 1=\frac{3 n(n+1)}{2}+n \end{align}$

Conclusion

Sigma and Pi notations are used whenever we try to deal with a sequence of numbers. It is very frequently used in the sequence and series chapter in the JEE syllabus. We have provided all the required formulas and theorems required for solving questions of this concept. We also discussed how to convert summation to multiplication along with formulas and properties of sigma function and sigma notation in statistics. Sigma and Pi notation, like other arithmetic notations, save a lot of paper and ink by allowing fairly complicated ideas to be expressed in a small amount of space. When working with sequences and series, the Sigma (summation) and Pi (product) notations allow you to define each term's value using the term number that describes all terms in the sum or product.