SUMMATION FORMULA

SUMMATION FORMULA

Edited By Komal Miglani | Updated on Jul 02, 2025 06:39 PM IST

Summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. If we add or subtract all the terms of a sequence we will get an expression, which is called a series. A series can be simply represented using summation, often known as sigma notation. In real life, we use Summation in mathematics and statistics to represent the sum of a series of numbers.

This Story also Contains
  1. Summation by Sigma(Σ) Operator
  2. Properties of Sigma Notation
  3. Solved Examples Based on Summation by Sigma Operator
SUMMATION FORMULA
SUMMATION FORMULA

In this article, we will cover the concept of Summation by Sigma Operator. This category falls under the broader category of Sequence and series, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main Exam (from 2013 to 2023), a total of 20 questions have been asked on this concept, including one in 2029, four in 2021, two in 2022, and twelve in 2023.

Summation by Sigma(Σ) Operator

The summation of each term of a sequence or a series can be represented in a compact form, called summation or sigma notation. This summation is represented by the Greek capital letter, Sigma (Σ).

For example,

$\\\mathrm{\sum_{n=1}^{n=10} n\;,\;it\;means \;the \;sum\;of\;n\;terms\;when\;n\;varies\;from\;1\;to\;10}\\\mathrm{\sum_{n=1}^{n=10} n=1+2+3+4+5+6+7+8+9+10}$

If we have the formula for the rth term i.e. $A_r$ of the series, we can put the sum of n terms of the series in the form of sigma notation as

$S_{n}= a_{1}+ a_{2}+--------+ a_{n}$ = $\sum^n_{r=1} A_r$

Here, $A_r$ is called the general term of the series.

Thus, the sum of n terms of A.P. whose rth term is $A_r$ = a+ (r-1)*d; where a is the first term and d is the common difference is given by

$S_n = \sum^n_{r=1} A_r =\sum ^n_{r=1} a+ (r-1)*d$

In fact, we can put the sum of any series in the sigma notation if the formula for its rth term is known.

Properties of Sigma Notation

$\\\mathrm{1.\;\;\sum_{r=1}^{n}T_r=T_1+T_2+T_3+.......+T_n,\;where,\;T_r\;is\;the\;general\;term\;of\;the\;series.}\\\\\mathrm{2.\;\;\sum_{r=1}^{n}\left ( T_r\pm T_r' \right )=\sum_{r=1}^{n}T_r\pm\sum_{r=1}^{n}T_r'\;\;(sigma\;\;operator\;is\;distributive\;\;over\;addition\;and\;subtraction)}\\\\\mathrm{3.\;\;\sum_{r=1}^{n}T_rT_r'\neq\left ( \sum_{r=1}^{n}T_r \right )\left ( \sum_{r=1}^{n}T_r' \right )\;\;(sigma\;\;operator\;is\;not\;distributive\;\;over\;multiplication)}\\\\\mathrm{4.\;\;\sum_{r=1}^{n}\frac{T_r}{T_r'}\;\neq\;\frac{\sum_{r=1}^{n}T_r}{\sum_{r=1}^{n}T_r'}\;\;(sigma\;\;operator\;is\;not\;distributive\;\;over\;division})\\\\\mathrm{5.\;\;\sum_{r=1}^{n}aT_r=a\sum_{r=1}^{n}T_r\;\;\;\;(a\;is\;constant)}\\\\\mathrm{6.\;\;\sum_{j=1}^{n}\sum_{i=1}^{n}T_iT_j=\left ( \sum_{i=1}^{n}T_i \right )\left ( \sum_{j=1}^{n}T_j \right )\;\;\;(here\;i\;and\;j\;are\;independent)}$


Solved Examples Based on Summation by Sigma Operator

Example 1: Let $<a_n>$ be a sequence such that $a_1+a_2+\ldots+a_a=\frac{n^2+3 n}{(n+1)(n+2)}$, If $28 \sum_{k=1}^{10} \frac{1}{a_k}=p_1 p_2 p_3 \ldots p_m$, where $p_1, p_2 \ldots \ldots \mathrm{P}_w$ are the first m prime numbers, then m is equal to [JEE MAINS 2023]

Solution

$\begin{aligned} & a_n=S_n-S_{n-1}=\frac{n^2+3 n}{(n+1)(1+2)}-\frac{(n-1)(n+2)}{n(n+1)} \\ & \Rightarrow a_n=\frac{4}{n(n+1)(1+2)} \\ & \Rightarrow 28 \sum_{k-1}^{10} \frac{1}{a_k}=28 \sum_{k=1}^{10} \frac{k(k+1)(k+2)}{4} \\ & =\frac{7}{4} \sum_{k=1}^{10}(k(k+1)(k+2)(k+3)-(k-1) k(k+1)(k+2) \\ & =\frac{7}{4} .10 .11 .12 .13=2.3 .5 .7 .11 .13 \\ & \text { So } m=6 \\ & \end{aligned}$

Hence, the answer is 6

Example 2: Let $\sum_{n=0}^{\infty} \frac{n^3((2 n) !)+(2 n-1)(n !)}{(n !)((2 n) !)}=a e+\frac{b}{e}+c$, where $\text { a, b, c } \in \mathbb{Z}$ and $\mathrm{e}=\sum_{n=0}^{\infty} \frac{1}{n !}$ Then $a^2-b+c$ is equal to : [JEE MAINS 2023]

Solution

$\begin{aligned} & \text { Let } \sum_{n=0}^{\infty} \frac{n^3((2 n) !)+(2 n-1) n !}{(n !)((2 n) !)} \\ & =\sum_{n=0}^{\infty} \frac{n^3(2 n) !}{n !(2 n) !}+\frac{(2 n-1) n !}{n !(2 n) !} \\ & =S_1+S_2 \end{aligned}$

$\text { Let } \begin{aligned} S_1 & =\sum_{n=0}^{\infty} \frac{n^3(2 n) !}{n !(2 n) !}=\sum_{n=0}^{\infty} \frac{n^3}{n !}=\sum_{n=1}^{\infty} \frac{n^2}{(n-1) !} \\ & =\sum_{n=1}^{\infty} \frac{n^2-1+1}{(n-1) !} \\ & =\sum_{n=2}^{\infty} \frac{(n+1)}{(n-2) !}+\sum_{n=1}^{\infty} \frac{1}{(n-1) !} \end{aligned}$

$\begin{aligned} & =\sum_{n=2}^{\infty} \frac{(n-2)+3}{(n-2) !}+\sum_{n=1}^{\infty} \frac{1}{(n-1) !} \\ & =\sum_{n=3}^{\infty} \frac{1}{(n-3) !}+3 \sum_{n=2}^{\infty} \frac{1}{(n-2) !}+\sum_{n=1}^{\infty} \frac{1}{(n-1) !} \\ & S_1=e+3 e+e=5 e \\ & \because S_2=\sum_{n=0}^{\infty} \frac{(2 n-1) n !}{n !(2 n) !} \\ & =\sum_{n=0}^{\infty} \frac{2 n-1}{(2 n) !} \end{aligned}$

$\begin{aligned} & =\sum_{n=1}^{\infty} \frac{1}{(2 n-1) !}-\sum_{n=0}^{\infty} \frac{1}{(2 n) !} \\ & =\left(\frac{1}{1 !}+\frac{1}{3 !}+\frac{1}{5 !}+\ldots\right)-\left(1+\frac{1}{2 !}+\frac{1}{4 !}+\ldots .\right) \\ & =-1+\frac{1}{1 !}-\frac{1}{2 !}+\frac{1}{3 !}-\frac{1}{4 !}+\frac{1}{5 !}-\ldots \\ & =-\left(1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}+\ldots .\right) \\ & =-\mathrm{e}^{-1} \end{aligned}$

$\mathrm{S}_1+\mathrm{S}_2=5 \mathrm{e}-\frac{1}{\mathrm{e}}=\mathrm{ae}+\frac{\mathrm{b}}{\mathrm{e}}+\mathrm{c}$

Compare both sides

$\begin{aligned} & \mathrm{a}=5, \mathrm{~b}=-1, \mathrm{c}=0 \\ & \mathrm{a}^2-\mathrm{b}+\mathrm{c}=25+1+0=26 \end{aligned}$

Hence, the answer is 26.

Example 3: Let f(x) be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in N$ if $f(1)=3$ and $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}(\mathrm{k})=3279$ then the value of n is. [JEE MAINS 2023]

Solution

$\begin{aligned} &\begin{aligned} & f(x+y)=f(x) \cdot f(y), x, y \in N \\ & f(2)=3^2 \end{aligned}\\ &\begin{aligned} f(3)=3^3 \quad & \therefore 3 \frac{\left[3^n-1\right]}{2}=3279 \\ & 3^n-1=1093 \times 2 \\ & 3^n-1=2186 \\ & 3^n=2187 \\ & n=7 \end{aligned} \end{aligned}$

Hence, the answer is 7

Example 4: Let $\mathrm{s}_1, \mathrm{~s}_2, \mathrm{~s}_3, \ldots \ldots, \mathrm{s}_{10}$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $1,2,3, \ldots, 10$ and the common differences are $1,3,5, \ldots \ldots \ldots, 19$ respectively. Then $\sum_{\mathrm{i}=1}^{10} \mathrm{~s}_{\mathrm{i}}$ is equal to [JEE MAINS 2023]

Solution

$\begin{aligned} & \mathrm{S}_{\mathrm{k}}=6(2 \mathrm{k}+(11)(2 \mathrm{k}-1)) \\ & \mathrm{S}_{\mathrm{k}}=6(2 \mathrm{k}+22 \mathrm{k}-11) \\ & \mathrm{S}_{\mathrm{k}}=144 \mathrm{k}-66 \\ \end{aligned}$

$\begin{aligned} & \sum_1^{10} \mathrm{~S}_{\mathrm{k}}=144 \sum_{\mathrm{k}=1}^{10} \mathrm{k}-66 \times 10 \\ & =144 \times \frac{10 \times 11}{2}-660 \\ & =7920-660 \\ & =7260 \end{aligned}$

Hence, the answer is 7260

Example 5: Let $[\alpha]$ denote the greatest integer $\leq \alpha$ . Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[\sqrt{120}]$ is equal to _________. [JEE MAINS 2023]

Solution

$\begin{aligned} & \mathrm{S}=[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}] \\ & {[\sqrt{1}] \rightarrow[\sqrt{3}]=1 \times 3} \\ & {[\sqrt{4}] \rightarrow[\sqrt{8}]=2 \times 5} \\ & {[\sqrt{9}] \rightarrow[\sqrt{15}]=3 \times 7} \\ & \vdots \\ & {[\sqrt{100}] \rightarrow[\sqrt{120}]=10 \times 21} \\ & \mathrm{~S}=1 \times 3+2 \times 5+3 \times 7+\ldots+10 \times 21 \\ & =\sum_{\mathrm{r}=1}^{10} \mathrm{r}(2 \mathrm{r}+1) \\ & =2 \sum_{\mathrm{r}=1}^{10} \mathrm{r}^2+\sum_{\mathrm{r}=1}^{10} \mathrm{r} \\ & =\frac{2 \times 10 \times 11 \times 21}{6}+\frac{10 \times 11}{2} \\ & =770+55 \\ & =825 \end{aligned}$

Hence, the answer is (825).

Frequently Asked Questions (FAQs)

1. What is summation?

The summation of each term of a sequence or a series can be represented in a compact form, called summation or sigma notation. This summation is represented by the Greek capital letter, Sigma (Σ).

2. How do you represent the sum of the n term of AP by sigma notation?

 The sum of n terms of A.P. whose rth term is  A_r = a+ (r-1)*d; where a is the first term and  d is the common difference is given by

S_n = \sum^n_{r=1} A_r =\sum ^n_{r=1} a+ (r-1)*d

3. Why is the summation formula useful in mathematics?
The summation formula is useful because it provides a concise way to represent long series of additions, simplifies complex calculations, and allows for easier manipulation and analysis of sequences and series in various mathematical and real-world applications.
4. What's the difference between Σ and Π in mathematical notation?
Σ (Sigma) represents summation, or addition of a series of terms. Π (Pi) represents the product of a series of terms. While Σ adds numbers, Π multiplies them.
5. Can a summation have an infinite upper bound?
Yes, a summation can have an infinite upper bound. This is denoted by using the infinity symbol (∞) as the upper limit. Such sums are called infinite series and may or may not converge to a finite value.
6. What does the term "index of summation" mean?
The index of summation is the variable used in the summation formula that typically changes with each term. It's usually denoted by a letter like i, j, or k, and its range is specified at the bottom and top of the Σ symbol.
7. How do you simplify a summation with a constant term?
When summing a constant term, you can multiply the constant by the number of terms. For example, Σ(i=1 to n) c = c * n, where c is a constant.
8. What is the summation formula for the sum of the first n positive integers?
The summation formula for the sum of the first n positive integers is: Σ(i=1 to n) i = n(n+1)/2. This is also known as the triangular number formula.
9. How do you prove a summation formula?
Summation formulas can be proved using various methods, including mathematical induction, comparing series, using known formulas, or geometric visualization. The choice of method often depends on the specific formula and its complexity.
10. How do you handle summations with multiple variables?
Summations with multiple variables are typically handled using nested sums or by defining a relationship between the variables. The specific approach depends on the problem and the relationship between the variables.
11. What is the telescoping series technique in summations?
The telescoping series technique involves rewriting terms in a sum so that most terms cancel out, leaving only a few terms. This is often used to simplify complex sums or to find closed forms for series.
12. What is the summation formula for the sum of cubes?
The summation formula for the sum of cubes is: Σ(i=1 to n) i³ = (n²(n+1)²)/4. This formula has applications in various areas of mathematics and physics.
13. How do you handle summations with absolute values?
Summations with absolute values often require case-by-case analysis. You may need to split the sum into positive and negative parts, or use properties of absolute values to simplify the expression before summing.
14. How do you simplify a double summation?
To simplify a double summation, you can sometimes change the order of summation, combine like terms, or use known formulas for inner or outer sums. The specific approach depends on the form of the summation.
15. How do you handle summations with logarithmic terms?
Summations with logarithmic terms often don't have simple closed forms. They may be approximated using integral approximations, handled using special functions like the polylogarithm, or manipulated using properties of logarithms.
16. What is the connection between summation and recurrence relations?
Summation formulas can often be derived from recurrence relations, and vice versa. Many sequences defined by recurrence relations have sums that can be expressed in closed form using summation notation.
17. How do you handle summations with trigonometric functions?
Summations with trigonometric functions often involve recognizing patterns, using trigonometric identities, or applying complex exponential representations. Some trigonometric sums have closed forms, while others may require approximation techniques.
18. How do you simplify a summation with a variable upper limit?
Simplifying a summation with a variable upper limit often involves expressing the sum as a function of the upper limit. This may require recognizing patterns, using known formulas, or proving a general formula by induction.
19. How do you handle summations with complex numbers?
Summations with complex numbers are handled similarly to real number summations, but may involve additional techniques from complex analysis. Properties of complex exponentials and roots of unity are often useful in simplifying these sums.
20. How do you handle summations with floor or ceiling functions?
Summations with floor or ceiling functions often require case-by-case analysis or grouping of terms. Sometimes, properties of these functions can be used to simplify the sum or convert it to a more manageable form.
21. What is the relationship between summation and generating functions?
Generating functions are formal power series where the coefficients represent a sequence. Summation notation is often used to define or manipulate generating functions, providing a powerful tool for analyzing sequences and series.
22. Can you have nested summations?
Yes, you can have nested summations. These are summations within summations, often used for multi-dimensional arrays or complex series. They are evaluated from the innermost sum to the outermost.
23. How does changing the order of summation affect the result?
For finite sums, changing the order of summation doesn't affect the result. However, for infinite series, changing the order can sometimes lead to different results or affect convergence.
24. Can you have a summation with a decreasing index?
Yes, you can have a summation with a decreasing index. This is often denoted by switching the upper and lower bounds and using a negative step. For example, Σ(i=n to 1) f(i) is equivalent to Σ(i=1 to n) f(n-i+1).
25. What is the summation formula for geometric sequences?
The summation formula for a geometric sequence is: Σ(i=0 to n) ar^i = a(1-r^(n+1))/(1-r), where 'a' is the first term and 'r' is the common ratio (r ≠ 1).
26. How does the summation formula relate to sequences?
The summation formula is used to find the sum of terms in a sequence. Each term in the sequence becomes a term in the summation, with the index typically corresponding to the position in the sequence.
27. What is the difference between Σ(i=1 to n) i and Σ(i=0 to n) i?
The main difference is the starting point. Σ(i=1 to n) i sums integers from 1 to n, while Σ(i=0 to n) i includes 0 and sums from 0 to n. The former equals n(n+1)/2, while the latter equals n(n+1)/2 + 0 = n(n+1)/2.
28. How do you simplify a summation of polynomials?
To simplify a summation of polynomials, you can split it into separate sums for each term of the polynomial using the linearity of summation. Then, simplify each sum separately using known formulas or techniques.
29. What is the summation formula for the sum of squares?
The summation formula for the sum of squares is: Σ(i=1 to n) i² = n(n+1)(2n+1)/6. This formula is useful in many statistical and mathematical applications.
30. What is the connection between summation and series?
A series is the sum of a sequence, and summation notation is used to represent series. The terms of the sequence become the terms of the summation, and the result of the summation is the sum of the series.
31. How do you find the average of a set of numbers using summation notation?
The average of a set of numbers can be found using summation notation as follows: (1/n) * Σ(i=1 to n) x_i, where x_i represents each number in the set and n is the total count of numbers.
32. How do you handle summations with factorial terms?
Summations with factorial terms often don't have simple closed forms. They are typically handled using special functions like the gamma function, or approximated using techniques from asymptotic analysis.
33. What is the difference between a partial sum and an infinite sum?
A partial sum is the sum of a finite number of terms in a series, while an infinite sum (if it exists) is the limit of the partial sums as the number of terms approaches infinity.
34. What is the summation formula for the harmonic series?
The harmonic series is Σ(n=1 to ∞) 1/n. This series doesn't have a closed-form sum as it diverges. However, the partial sum can be approximated by ln(n) + γ, where γ is the Euler-Mascheroni constant.
35. How do you handle summations with alternating signs?
Summations with alternating signs can often be simplified by grouping terms, using properties of even and odd functions, or applying techniques for alternating series. The approach depends on the specific form of the summation.
36. What is the relationship between summation and finite differences?
Summation and finite differences are inverse operations, similar to integration and differentiation. The sum of finite differences of a sequence gives the original sequence (up to a constant).
37. How do you simplify a summation of exponential terms?
Simplifying a summation of exponential terms often involves recognizing geometric series, using properties of exponents, or applying special techniques for exponential sums. The method depends on the specific form of the exponential terms.
38. What is the summation formula for triangular numbers?
The summation formula for triangular numbers is: Σ(i=1 to n) i = n(n+1)/2. This formula represents the sum of the first n positive integers and has many applications in combinatorics and other areas of mathematics.
39. How do you simplify a summation of rational functions?
Simplifying a summation of rational functions may involve partial fraction decomposition, telescoping series techniques, or recognizing patterns in the numerators and denominators. The specific approach depends on the form of the rational function.
40. What is the summation formula for the sum of even numbers?
The summation formula for the sum of the first n even numbers is: Σ(i=1 to n) 2i = n(n+1). This can be derived from the formula for the sum of the first n positive integers.
41. What is the relationship between summation and combinatorics?
Summation notation is frequently used in combinatorics to represent counts of combinations or permutations. Many combinatorial identities can be expressed and proved using summation formulas.
42. What is the summation formula for the sum of odd numbers?
The summation formula for the sum of the first n odd numbers is: Σ(i=1 to n) (2i-1) = n². This formula represents the square numbers and has geometric interpretations.
43. What is the connection between summation and probability theory?
Summation is fundamental in probability theory, used to calculate expected values, variances, and probabilities of discrete random variables. Many probability distributions are defined using summation notation.
44. How do you simplify a summation with a step size other than 1?
To simplify a summation with a step size other than 1, you can often make a change of variable to convert it to a sum with step size 1. Alternatively, you may need to adapt known formulas or derive new ones for the specific step size.
45. What is the summation formula for the sum of reciprocals of triangular numbers?
The sum of reciprocals of triangular numbers is: Σ(n=1 to ∞) 2/(n(n+1)) = 2. This is an example of a telescoping series and demonstrates that some infinite series of positive terms can converge to a finite sum.
46. How do you simplify a summation involving binomial coefficients?
Simplifying summations with binomial coefficients often involves recognizing combinatorial identities, using properties of Pascal's triangle, or applying techniques from generating functions. The specific approach depends on the form of the summation.
47. What is the connection between summation and numerical integration?
Summation forms the basis for many numerical integration techniques, such as the rectangle method, trapezoidal rule, and Simpson's rule. These methods approximate integrals by summing the areas of many small shapes that approximate the area under a curve.

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