Sequences and Series Class 11 Notes CBSE Maths Chapter 8 (Free PDF Download)

Section–A (1 Mark Questions)

1. If the sum of n terms of an A.P. is given by Sn=3n+2n2 then find the common difference of the A.P.

Ans. Given, $S_n=3 n+2 n^2$

$$ \begin{aligned} & S_1=3(1)+2(1)^2=5=a_1 \\ & S_2=3(2)+2(2)^2=14=a_1+a_2 \\ & \therefore S_2-S_1=9=a_2 \\ & \therefore d=a_2-a_1=9-5=4 . \end{aligned} $$

2. If the third term of G.P. is 4, then what is the product of its first 5 terms.

Ans. Let $a$ and $r$ be the first term and common ratio of G.P., respectively.

Given that the third term is 4 .

$$\therefore a r^2=4$$

Product of first 5 terms

$$=a \cdot a r \cdot a r^2 \cdot a r^3 \cdot a r^4=a^5 r^{10}=\left(a r^2\right)^5=4^5\text {. }$$

3. The 17th term from the end of A.P. -36,-31,-26,.....79 is ________. 

Ans. Here, $a=-36$ and $d=-31-(-36)=5$

$$\begin{aligned}& l=79 \\& \therefore 17^{\text {th }} \text { term from the end }=l-(n-1) d \\& =79-(17-1)(5)=79-80=-1 .\end{aligned}$$

4. Sum of the series $3+1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+....$ n terms, is ________.

Ans. The formula for the summation of $n$ terms of an geometric series is $S_n=\frac{a\left(1-r^n\right)}{1-r}$, where $a$ is the first term in the series and $r$ is the rate of change between successive terms.

Here $a=3$ and $r=\frac{1}{3}$

$$S_n=\frac{3\left(1-\left(\frac{1}{3}\right)^n\right)}{1-\left(\frac{1}{3}\right)}=\frac{9}{}\left(1-\left(\frac{1}{3}\right)^n\right) \text {. }$$

5. The first two terms of the sequence defined by a1=3 and an=3an-1 + 2 for all n>1 ________. 

Ans. Given: $a_1=3$ and $a_n=3 a_{n-1}+2$, for all $n>1$

When $n=2$ :

$$\begin{aligned}& a_2=3 a_{2-1}=3 a_1+2=3(3)+2 \\& =9+2=11 .\end{aligned}$$

Section-B (2 Marks Questions)

6. If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then prove that the 22nd term of the A.P. is zero.

Ans. Let the first term and common difference of given A.P. be $a$ and $d$, respectively.

It is given that $9 a_9=13 a_{13}$

$$ \begin{aligned} & \Rightarrow 9(a+8 d)=13(a+12 d) \\ & \Rightarrow 9 a+72 d=13 a+156 d \\ & \Rightarrow 4 a+84 d=0 \\ & \Rightarrow(a+21 d)=0 \\ & \Rightarrow a_{22}=0 . \end{aligned} $$

7. If a,b and c are in G.P., then find the value of $\frac{a-b}{b-c}$ .

Ans. Given that, $a, b$ and $c$ are in G.P.

$\Rightarrow b=a r$ and $c=a r^2$, where $r$ is the common ratio.

$$\Rightarrow \frac{a-b}{b-c}=\frac{a-a r}{a r-a r^2}=\frac{a(1-r)}{ar(1-r)}=\frac{1}{r}=\frac{a}{b} o r \frac{b}{c}$$

8. The sum of terms equidistant from the beginning and end in an A.P. is equal to ________.

Ans. Let $a$ be the first term and $d$ be the common difference of the A.P.

$a_r=r^{4 t}$ term from the beginning $=a+(r-1) d$

$a_r^{\prime}=r^{\text {di }}$ term from the end $=(a+(n-1) d)+(r-1)(-d)$

(as first term is $a_n=a+(n-1) d$ and common difference is ' $-d^{\prime}$ )

Now,

$$a_r+a^{\prime} r=a+(r-1) d+(a+(n-1) d)+(r-1)(-d)$$

$=2 a+(n-1) d$, which is independent of ' $r$ '

Thus, sum of the terms equidistant from the beginning and end in an A.P. is constant.

9. A man saved Rs. 66000 in 20 years. In each succeeding year after the first year, he saved Rs. 200 more than what he saved in the previous year. How much did he save in the first year?

Ans. Let us assume that the man saved Rs. $a$ in the first year.

In each succeeding year, an increment of Rs. 200 is made.

So, it forms an A.P. whose first term $=a$, common difference, $d=200$ and $n=20$

$$ \begin{aligned} & \therefore S_{20}=\frac{20}{2}[2 a+(20-1) d] \\ & \Rightarrow 66000=10[2 a+19 \times 200] \\ & \Rightarrow 6600=2 a+19 \times 200 \\ & \Rightarrow 2 a=2800 \\ & \therefore a=1400 . \end{aligned} $$

10. The sum of interior angles of a triangle is $180^{\circ}$. Show that the sum of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21-sided polygon.

Ans. We know that the sum of interior angles of a polygon of side $n$ is $(n-2) \times 180^{\circ}$.

Let $a_n=(n-2) \times 180^{\circ}$

Since $a_n$ is linear in $n$, it is $n^{\text {,h }}$ term of some A.P.

$$a_3=(3-2) \times 180^{\circ}=180^{\circ}$$

Common difference, $d=180^{\circ}$

Sum of the interior angles for a 21-sided polygon is:

$$a_{21}=(21-2) \times 180^{\circ}=3420^{\circ} \text {. }$$

11. Find the $r^{th}$ term of an A.P., whose sum of first n terms is $2n+3n^{2}$ .

Ans. Sum of $n$ terms of A.P., $S_n=2 n+3 n^2$

$$a_n=S_n-S_{n-1}$$

$$\begin{aligned}& =\left(2 n+3 n^2\right)-\left[2(n-1)+3(n-1)^2\right] \\& =[2 n-2(n-1)]+\left[3 n^2-3(n-1)^2\right] \\& =2(n-n+1)+3(n-n+1)(n+n-1) \\& =2+3(2 n-1) \\& =6 n-1 \\& \therefore r^{\text {sn }} \text { term } a_r=6 r-1 .\end{aligned}$$

12. Show that the products of the corresponding terms of the sequences $a,ar,ar^{2},....ar^{n-1}$ and $A,AR,AR^{2},....AR^{n-1}$ form a G.P., and find the common ratio.

Ans. It has to be proved that the sequence, $a A, \operatorname{ar} A R, a r^2 A R^2, \ldots a r^{n-1} A R^{n-1}$, forms G.P

$\dfrac{\text { Second term }}{\text { First term }}=\dfrac{a r A R}{a A}=r R$

$\dfrac{\text { Third term }}{\text { Second term }}=\dfrac{a r^2 A R^2}{a r A R}=r R$

Thus, the above sequenee forms a G.P. and the common ratio is $r R$.

13. If the sum of n terms of an A.P. is $(pn+qn^{2})$, where p and q are constants, find the common difference.

Ans. It is known that,

$$S_n=\frac{n}{2}[2 a+(n-1) d]$$

According to the given condition,

$$\begin{aligned}& \frac{n}{2}[2 a+(n-1) d]=p n+q n^2 \\& \Rightarrow \frac{n}{2}[2 a+n d-d]=p n+q n^2 \\& \Rightarrow n a+n^2 \frac{d}{2}-n \cdot \frac{d}{2}=p n+qn^2\end{aligned}$$

Comparing the coefficients of $n^2$ on both sides, we obtain

$$\frac{d}{2}=q \Rightarrow d=2 q$$

Thus, the common difference of the A.P. is $2 q$.

PDF Summary - Class 11 Maths Sequences and Series Notes (Chapter 8)

1. Definition:

Any function with domain as a set of natural numbers is called sequence.

Real sequence: Sequence with range as subset of real numbers.

Series:

For example:-

If \[{a_1}\;,\;{a_2}\;,{a_3}\;,\;...............\;,\;{a_n}\]is a sequence, then \[{a_1}\; + \;{a_2}\; + \;{a_3}\; + \;...............\; + \;{a_n}\]  is a series.

Progression: When terms of a sequence follow a certain pattern. 

But it is not always necessary that terms of sequence follow a certain pattern.

1.1 Arithmetic Progression (AP):

An arithmetic progression is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed number.

If this fixed number is positive, then it is an increasing AP and if this fixed number is negative, then it is a decreasing AP.

This fixed term is called common difference and is usually represented by ‘d’.

Let ‘a’ be the first term of an AP.

Nth term of an AP: \[{t_n}\; = \;a\; + \left( {n\; - \;1} \right)d\;,\;{\text{where}}\;\;d\; = \;{a_n} - \;{a_{n - 1}}\]

Sum of first N terms of an AP: \[{S_n}\; = \;\dfrac{n}{2}\left[ {a\; + \;\left( {n\; - \;1} \right)d} \right]\; = \;\dfrac{n}{2}\left[ {a\; + \;l} \right]\;{\text{where}}\;{\text{,}}\;l\;{\text{is}}\;{\text{last}}\;{\text{term}}\;{\text{of}}\;{\text{an}}\;{\text{AP}}\].

Properties of an AP:

• Increasing, Decreasing, Multiplying and dividing each term of an AP by a non-zero constant results into an AP.

• 3 numbers in an AP: \[a\; - \;d\;,\;a\;,\;a\; + \;d\]

4 numbers in an AP: \[a\; - \;3d\;,\;a\; - \;d\;,\;a\; + \;d\;,\;a\; + \;3d\]

5 numbers in an AP: \[a\; - \;2d\;,\;a\; - \;d\;,\;a\;,\;a\; + \;d\;,\;a\; + \;2d\]

6 numbers in an AP: \[a\; - \;5d\;,\;a\; - \;3d\;,\;a\; - \;d\;,\;a\; + \;d\;,\;a\; + \;3d\;,\;a\; + \;5d\]

• An AP can have zero, positive or negative common difference.

•  The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms.

• Any term of an AP (except the first) is equal to half the sum of terms which are equidistant from it.

\[ \Rightarrow \;\;{a_n}\; = \;\dfrac{1}{2}\left( {{a_{n\; - \;k}}\; + \;{a_{n\; + \;k}}} \right)\;\;,\;k\; < \;n\]

• \[{t_r}\; = \;{S_r}\; - \;{S_{r\; - \;1}}\]

• If three numbers are in AP : a, b, c are in AP \[ \Rightarrow \;2b\; = \;a\; + \;c\]

• Nth term of an AP is a linear expression in n: \[An\; + \;B\]where A is the common difference of an AP.

1.2 Geometric Progression (GP):

It is a sequence in which each term is obtained by multiplying the preceding term by a fixed number (which is constant) called common ratio. First term of GP is non zero.

Common ratio can be obtained by dividing a term by its consecutive preceding term.

If ‘a’ is the first term and ‘r’ is the common ratio then,

GP is \[a\;,ar\;,\;a{r^2}\;,\;a{r^3}\;,\;a{r^4}\;,\;.\;.\;.\;.\;.\;.\;.\;\]

Nth term of a GP: \[{t_n}\; = \;a{r^{n - 1}}\]

Sum of first N terms of a GP: \[{s_n}\; = \;\dfrac{{a\left( {1\; - \;{r^n}} \right)}}{{\left( {1\; - \;r} \right)}}\;,\;r\; \ne \;1\]

Sum of infinite GP when \[|r|\; < \;1\;\;\& \;\;n\; \to \;\infty \]

\[|r|\; < \;1\;\; \Rightarrow \;{r^n}\; \to \;0\; \Rightarrow \;{S_\infty }\; = \;\dfrac{a}{{1\; - \;r}}\]

Properties of a GP:

• Multiplying and dividing each term of a GP by a non- zero constant results into a GP.

• Reciprocal of terms of GP is also GP.

• 3 consecutive terms in GP: \[\dfrac{a}{r}\;,\;a\;,\;ar\]

4 consecutive terms in GP: \[\dfrac{a}{{{r^2}}}\;,\;\dfrac{a}{r}\;,\;ar\;,\;a{r^2}\]

• If three numbers are in GP : a, b, c are in GP \[ \Rightarrow \;{b^2}\; = \;ac\]

• Each term of a GP raised to the same power also forms a G.P.

• Choosing terms of GP at regular intervals also forms a GP.

• The product of the terms equidistant from the beginning and the last is always same and is equal to the product of the first and the last term for a finite GP.

• If \[{a_1}\;,\;{a_2}\;,{a_3}\;,\;...............\;,\;{a_n}\] forms GP with non-zero and non-negative terms then\[\log {a_1}\;,\;\log {a_2}\;\log ,{a_3}\;,\;...............\;,\;\log {a_n}\] are in GP or vice versa.

2. Means:

2.1 Arithmetic Mean

When three terms are in AP, the middle term is called AM between the other two.

If a, b, c are in AP, b is AM between a and c.

If n positive terms \[{a_1}\;,\;{a_2}\;,{a_3}\;,\;...............\;,\;{a_n}\] are in AP, then AM is:

\[A\; = \;\dfrac{{{a_1}\; + \;{a_2}\; + \;{a_3}\; + \;.......\; + \;{a_n}}}{n}\]

2.2 n-Arithmetic Means Between Two Numbers

If a, b are two numbers and \[a\;,\;{a_1}\;,\;{a_2}\;,{a_3}\;,\;...............\;,\;{a_n}\;,\;b\]are in an AP, then

\[{a_1}\;,\;{a_2}\;,{a_3}\;,\;...............\;,\;{a_n}\] are n AM’s between a and b.

\[{A_1}\; = \;a\; + \;d\;,\;{A_2}\; = \;a\; + \;2d\;,\;...........\;,\;{A_n}\; = \;a\; + \;nd\], where \[d\; = \;\dfrac{{b\; - \;a}}{{n\; + \;1}}\]

NOTE: Sum of n AM’s inserted between a and b is equal to n times a single AM between a and b.\[\Rightarrow \; \sum\limits_{r\; = \;1}^n {{A_r}}\; = \;nA\]

2.3 Geometric Mean

If \[a,{\text{ }}b,{\text{ }}c\]are in GP, then b is called GM between a and c.

So, \[{b^2}\; = \;ac\;\;or\;\;b\; = \;\sqrt {ac} \;;\;a > 0\;,\;b > 0\]

2.4 n-Geometric Means between two numbers 

If a, b are two numbers and \[a\;,\;{G_1}\;,\;{G_2}\;,{G_3}\;,\;...............\;,\;{G_n}\;,\;b\]are in a GP, then

\[{G_1}\;,\;{G_2}\;,{G_3}\;,\;...............\;,\;{G_n}\] are n GM’s between a and b.

\[{G_1}\; = \;ar\;,\;{G_2}\; = \;a{r^2}\;,..........,\;{G_n}\; = \;a{r^{n\; - \;1}}\], where \[r\; = \;{\left( {\dfrac{b}{a}} \right)^{\dfrac{1}{{n\; + \;1}}}}\]

NOTE: Product of n GM’s inserted between a and b is equal to nth power of single GM between a and b. \[ \Rightarrow \;{\prod\limits_{r\; = \;1}^n {{G_r}\; = \;\left( G \right)} ^n}\]

2.5 Arithmetic, Geometric and Harmonic means between two given numbers

Let A, G and H be the arithmetic, geometric and harmonic mean between two integers numbers a and b.

\[ \Rightarrow \;A\; = \;\dfrac{{a\; + \;b}}{2}\;,\;G\; = \;\sqrt {ab} \;,\;H\; = \;\dfrac{{2ab}}{{a\; + \;b}}\]

The three means have following three properties:

1. \[A\; \geqslant \;G\; \geqslant \;H\]

2. \[{G^2}\; = \;AH\] which means that A, G, H forms a GP.

3. Equation \[{x^2}\; - \;2Ax\; + \;{G^2}\; = \;0\] have a and b as its roots.

4. If A, G, H are corresponding means between three given numbers a, b and c, then the equation having a, b, c as its roots is \[{x^3}\; - \;3A{x^2}\; + \;\dfrac{{3{G^2}}}{H}x\; - \;{G^3}\; = \;0\]

NOTE: Some important properties of Arithmetic & Geometric Means between two quantities:

1. If A and G are arithmetic and geometric mean between a and b then Quadratic equation \[{x^2}\; - \;2Ax\; + \;{G^2}\; = \;0\]has a and b as its roots.

2. If A and G are AM and GM between two numbers a and b, then

\[a\; = \;A\; + \;\sqrt {{A^2}\; - \;{G^2}} \], \[b\; = \;A\; - \;\sqrt {{A^2}\; - \;{G^2}} \]

3. Sigma Notations:

3.1 Theorems

(i) \[\sum\limits_{r\; = \;1}^n {({a_r}\; + \;{b_r})\; = \;\sum\limits_{r\; = \;1}^n {{a_r}\; + \;\sum\limits_{r\; = \;1}^n {{b_r}} } } \]

(ii) \[\sum\limits_{r\, = \;1}^n {ka\; = \;k\;\sum\limits_{r\, = \;1}^n {{a_r}} } \]

(iii) \[\sum\limits_{r\, = \;1}^n k \; = \;nk\]

4. Sum of n Terms of Some Special Sequences

4.1 Sum of first n natural numbers

\[\sum\limits_{k\, = \;1}^n k \; = \;1\; + \;2\; + \;3\; + \;.......\; + \;n\; = \;\dfrac{{n\left( {n\; + \;1} \right)}}{2}\]

4.2 Sum of squares of  first n natural numbers

\[\sum\limits_{k\, = \;1}^n {{k^2}} \; = \;{1^2}\; + \;{2^2}\; + \;{3^2}\; + \;.......\; + \;{n^2}\; = \;\dfrac{{n\left( {n\; + \;1} \right)\left( {2n\; + \;1} \right)}}{6}\]

4.3 Sum of cubes of  first n natural numbers 

\[\sum\limits_{k\, = \;1}^n {{k^3}} \; = \;{1^3}\; + \;{2^3}\; + \;{3^3}\; + \;.......\; + \;{n^3}\; = \;{\left[ {\dfrac{{n\left( {n\; + \;1} \right)}}{2}} \right]^2}\; = \;{\left[ {\sum\limits_{k\, = \;1}^n k } \right]^2}\]

5. Arithmetico-Geometric series

An arithmetic-geometric progression (A.G.P.) is a progression in which each term can be represented as the product of the terms of an arithmetic progression (AP) and a geometric progression (GP).

\[AP\;:\;1\;,\;3\;,\;5\;,\;..........\] and \[GP\;;\;1\;,\;x\;,\;{x^2}\;,........\]

\[ \Rightarrow \;\;AGP\;:\;1\;,\;3x\;,\;5{x^2},........\]

5.1 Sum of n terms of an Arithmetico-Geometric Series

${{\text{S}}_n} = {\text{a}} + ({\text{a}} + {\text{d}}){\text{r}} + ({\text{a}} + 2\;{\text{d}}){{\text{r}}^2} +  \ldots  \ldots  + $ $[a + (n - 1)d]{r^{n - 1}}$

then ${S_n} = \dfrac{a}{{1 - r}} + \dfrac{{dr\left( {1 - {r^{n - 1}}} \right)}}{{{{(1 - r)}^2}}} - \dfrac{{[a + (n - 1)d]{r^n}}}{{1 - r}},r \ne 1$

5.2 Sum to Infinity

If $|r| < 1{\text{ \& }}\;n \to \infty $, then $\mathop {\lim }\limits_{n \to \infty }  = 0.\;{S_\infty } = \dfrac{a}{{1 - r}} + \dfrac{{dr}}{{{{(1 - r)}^2}}}$.

6. Harmonic Progression (HP) 

A sequence, reciprocal of whose terms forms an AP is called HP.

If the sequence ${a_1},{a_2},{a_3}, \ldots  \ldots  \ldots  \ldots  \ldots ,{a_n}$ is an HP, then

$\dfrac{1}{{{a_1}}}\;,\;\dfrac{1}{{{a_2}}}\;,\;\dfrac{1}{{{a_3}}}\;,\;.......\;,\;\dfrac{1}{{{a_n}}}$ is an AP or vice versa. There is no formula for the sum of the $n$ terms of an HP. For HP with first terms is a and second term is $b$, then ${n^{{\text{th }}}}$ term is ${t_n} = \dfrac{{ab}}{{b + (n - 1)(a - b)}}$

If \[a,\;b,\;c\] are in ${\text{HP}} \Rightarrow {\text{b}} = \dfrac{{2{\text{ac}}}}{{{\text{a}} + {\text{c}}}}$ or $\dfrac{{\text{a}}}{{\text{c}}} = \dfrac{{{\text{a}} - {\text{b}}}}{{{\text{b}} - {\text{c}}}}$.

7. Harmonic Mean

If \[a,\;b,\;c\] are in HP then, \[b\] is the HM between \[a\;\& \;c\] \[ \Rightarrow \;b\; = \;\dfrac{{2ac}}{{a\; + \;c}}\]. 

Sequence and Series Class 11 Notes

Preparing from CBSE Sequence and Series Notes helps students to understand the important topics such as A.P, G.P, harmonic progressions, the arithmetic-geometric mean, and harmonic mean. These notes help students to get a good score in examinations. Topics are explained in very easy language which helps the students to understand and revise syllabus with almost no time in  Sequences and Series Revision Notes. Students can solve any MCQs and Subjective question paper, once they are thorough with the notes. So students are advised to study Class 11 Maths Chapter 8 Notes without any confusion. Let’s look at the topics covered in these notes. 

• Meaning of Sequence

• What is a sequence in Math?

• Finite Sequence

• Infinite Sequence

• Types of Sequence

• Arithmetic Sequence

• Geometric Sequence

• Fibonacci Sequence

• Meaning of Series

• Notation of Series

• Finite and Infinite Series

• Types of Series

• Arithmetic Series

• Geometric Series

• Meaning of Geometric Progression (G.P.)

• Meaning of Arithmetic Progression (A.P.)

• Arithmetic Mean

• Geometric Mean

• Relation between A.M. and G.M.

• Special Series

• Sum to n terms of Special Series

Meaning of Sequence

A sequence is nothing but a group of objects that follow some particular pattern. If we have some objects which are listed in some kind order so that it has 1st term, 2nd term and so on, then it is a sequence.

What is a Sequence in Math?

In Mathematics, it is defined as a group of numbers which are in an ordered form which follows a particular pattern is called Sequence. There are Finite Sequences and Infinite Sequences. The sequence which has a finite number of terms (Limited terms) is called Finite Sequence. The sequence that has an unlimited number of terms (Infinite terms) is called Infinite Sequence.

Types of Sequence

There are 3 types of sequences:

• Arithmetic Sequence

• Geometric Sequence

• Fibonacci Sequence

Arithmetic Sequence

In any sequence, if the difference between every successive term is a constant then it is defined as Arithmetic Sequence. It can be in ascending or descending order, but it has to be according to a constant number.

Geometric Sequence

In any sequence, if the ratio between each successive term is constant then it is known as Geometric Sequence. It can be in ascending or descending order according to the constant ratio.

Like we discussed a few topics above, Class 12 Maths, Chapter 8 Sequences and Series is a difficult subject with many problems and concepts. Many of the definitions are thoroughly clarified. As a result, learning all of these will require some extra effort, and students will need to keep revising and practising in order to completely master the subject. Though students may not have enough time to prepare notes on their own, we at Vedantu provide well-organized CBSE Class 11 Maths Notes Chapter 8 Sequences and Series that will assist them in their examination preparation as well as increase their interest in the concepts. Refer to the free PDF of CBSE Sequence and Series Notes for the complete notes.

Tips to Prepare for Exams Using CBSE Sequence and Series Notes

• You must complete the previous year's questions after you have completed the concepts and numerical. With the previous year's problems, you'll be able to see just where you're missing and how to progress accordingly.

• To improve your pace and accuracy, take online mock tests on a regular basis. This activity will be particularly beneficial in JEE Mains.

• Understand your strengths and weaknesses, and work to improve both.

• If you notice any questions that seem to be critical when practicing, make a note of them. You must solve the question again later when revising this chapter; this will help you brush up on your concepts.

• For this chapter, you should make a small formula notebook/flashcards and revise them weekly to keep them fresh in your mind.

Conclusion

The Sequence and Series Class 11 Notes prepared by Vedantu is helpful for students to score good marks in their board exams. These solutions are prepared based on important questions from the NCERT curriculum by the top faculty of Vedantu. The practice problems provided in CBSE Sequence and Series Notes will help students to revise the concepts and ace their exams. The solutions and concepts are prepared by experts to provide top-notch learning content to students. Experts have done a lot of research on the preparation of solutions to provide a unique and fun learning experience to students.