Given:

• The number of trees planted by a section of a class is equal to the class number.
• There are 12 classes in total (Class I to Class XII).
• Each class has 3 sections.

To Find:

The total number of trees planted by all the students of all classes.

Step 1: Determining the number of trees planted by each class

Let $n$ be the class number, where $n \in \{1, 2, 3, \dots, 12\}$.

Since there are 3 sections for each class, the number of trees planted by a specific class $n$ is given by:

$T_n = 3 \times n$

Calculating the trees for each class:

• Class I: $3 \times 1 = 3$
• Class II: $3 \times 2 = 6$
• Class III: $3 \times 3 = 9$
• ...
• Class XII: $3 \times 12 = 36$

Step 2: Identifying the Arithmetic Progression (AP)

The sequence of the total trees planted by each class is: $3, 6, 9, \dots, 36$.

This sequence forms an Arithmetic Progression where:

• First term ($a$) = $3$
• Common difference ($d$) = $6 - 3 = 3$
• Number of terms ($n$) = $12$
• Last term ($l$ or $a_{12}$) = $36$

Step 3: Calculating the sum of the Arithmetic Progression

To find the total number of trees, we calculate the sum of the first $n$ terms of the AP using the formula:

$S_n = \frac{n}{2} (a + l)$

[Where $S_n$ is the sum of $n$ terms, $a$ is the first term, and $l$ is the last term.]

Substituting the known values into the formula:

$S_{12} = \frac{12}{2} (3 + 36)$

Step 4: Performing the arithmetic operations

$S_{12} = 6 \times (39)$

$S_{12} = 234$

[Calculation breakdown: $6 \times 30 = 180$ and $6 \times 9 = 54$. Adding these: $180 + 54 = 234$.]

Final Answer: The total number of trees planted by the students is 234.

Given:

Principal amount ($P$) = ₹ $10000$

Rate of interest ($r$) = $8\%$ per annum

The interest is compounded annually.

To Find:

Whether the sequence of amounts at the end of each year forms an Arithmetic Progression (AP).

Step 1: Understanding the Formula for Compound Interest

The amount ($A$) after $n$ years with compound interest is given by the formula:

$A = P \left(1 + \frac{r}{100}\right)^n$

Where:

• $P = 10000$
• $r = 8$
• $n$ is the number of years ($n = 1, 2, 3, \dots$)

Step 2: Calculating the amount for consecutive years

Let $a_1, a_2, a_3, \dots$ be the amount in the account at the end of the 1st, 2nd, and 3rd year respectively.

For $n = 1$ (Amount at the end of 1st year):

$a_1 = 10000 \left(1 + \frac{8}{100}\right)^1 = 10000(1.08) = 10800$

For $n = 2$ (Amount at the end of 2nd year):

$a_2 = 10000 \left(1 + \frac{8}{100}\right)^2 = 10000(1.08)^2 = 10000(1.1664) = 11664$

For $n = 3$ (Amount at the end of 3rd year):

$a_3 = 10000 \left(1 + \frac{8}{100}\right)^3 = 10000(1.259712) = 12597.12$

Step 3: Checking for Arithmetic Progression

A sequence is an Arithmetic Progression if the difference between consecutive terms is constant (i.e., $a_{n+1} - a_n = d$, where $d$ is the common difference).

Calculate the first difference ($d_1$):

$d_1 = a_2 - a_1 = 11664 - 10800 = 864$

Calculate the second difference ($d_2$):

$d_2 = a_3 - a_2 = 12597.12 - 11664 = 933.12$

Step 4: Conclusion

Since $d_1 \neq d_2$ ($864 \neq 933.12$), the difference between consecutive terms is not constant.

[By definition, a sequence is an AP if and only if the common difference is constant for all terms.]

Final Answer: The list of numbers does not form an Arithmetic Progression because the difference between consecutive terms is not constant.

Given: An Arithmetic Progression (AP) sequence: $3, 8, 13, \dots, 253$.

To Find: The $20^{th}$ term from the end (last term) of the given AP.

Step 1: Identify the parameters of the given AP.

The sequence is $3, 8, 13, \dots, 253$.

First term ($a$) = $3$.

Common difference ($d$) = $a_2 - a_1 = 8 - 3 = 5$.

Last term ($l$) = $253$.

Step 2: Formulate the strategy to find the $n^{th}$ term from the end.

To find the $n^{th}$ term from the end of an AP, we can reverse the sequence. The new AP will have the last term of the original AP as its first term, and the common difference will be the negative of the original common difference ($-d$).

Let the reversed AP be: $253, 248, 243, \dots, 3$.

For this reversed AP:

New first term ($a'$) = $253$.

New common difference ($d'$) = $-5$.

We need to find the $20^{th}$ term ($n = 20$).

Step 3: Apply the formula for the $n^{th}$ term of an AP.

The formula for the $n^{th}$ term of an AP is given by:

$a_n = a' + (n - 1)d'$

[Where $a_n$ is the $n^{th}$ term, $a'$ is the first term, $n$ is the position, and $d'$ is the common difference.]

Step 4: Substitute the values into the formula.

$a_{20} = 253 + (20 - 1)(-5)$

$a_{20} = 253 + (19)(-5)$

[Performing the multiplication: $19 \times -5 = -95$]

$a_{20} = 253 - 95$

Step 5: Perform the final subtraction.

$a_{20} = 158$

Final Answer: The 20th term from the last term of the AP is 158.

Given: An Arithmetic Progression (AP) sequence: $10, 7, 4, \dots$

To Find: The $30^{th}$ term of the given AP.

Step 1: Identify the parameters of the Arithmetic Progression.

An Arithmetic Progression is a sequence of numbers such that the difference between consecutive terms is constant. This constant is known as the common difference ($d$).

The first term ($a$) is the first number in the sequence:
$a = 10$

The common difference ($d$) is calculated by subtracting the first term from the second term:
$d = a_2 - a_1$
$d = 7 - 10$
$d = -3$

Step 2: State the formula for the $n^{th}$ term of an AP.

The general formula to find the $n^{th}$ term ($a_n$) of an Arithmetic Progression is:
$a_n = a + (n - 1)d$
Where:
$a_n$ = the $n^{th}$ term to be found
$a$ = the first term
$n$ = the position of the term
$d$ = the common difference

Step 3: Substitute the known values into the formula.

We are looking for the $30^{th}$ term, so $n = 30$.
Substituting $a = 10$, $d = -3$, and $n = 30$ into the formula:

$a_{30} = 10 + (30 - 1)(-3)$

Step 4: Perform the arithmetic calculations.

First, solve the expression inside the parentheses:
$a_{30} = 10 + (29)(-3)$

Next, perform the multiplication:
$29 \times -3 = -87$

Finally, perform the addition:
$a_{30} = 10 - 87$
$a_{30} = -77$

Justification: By applying the standard formula for the $n^{th}$ term of an arithmetic progression, we have determined that the sequence decreases by $3$ at each step. Starting from $10$, after $29$ steps of decreasing by $3$, the value reaches $-77$.

Final Answer: The 30th term of the AP is -77.

Given: An Arithmetic Progression (AP) series: $7, 13, 19, \dots, 205$.

To find: The number of terms ($n$) in the given AP.

Step 1: Identify the parameters of the Arithmetic Progression.
An Arithmetic Progression is defined by its first term ($a$) and its common difference ($d$).
- The first term ($a$) is the first number in the sequence: $a = 7$.
- The second term ($a_2$) is $13$.
- The common difference ($d$) is calculated as the difference between any two consecutive terms: $d = a_2 - a_1$.
$d = 13 - 7 = 6$.
- The last term ($a_n$ or $l$) is given as $205$.

Step 2: State the relevant formula.
The formula for the $n^{th}$ term of an Arithmetic Progression is given by:
$a_n = a + (n - 1)d$
Where:
$a_n$ = the $n^{th}$ term
$a$ = the first term
$n$ = the number of terms
$d$ = the common difference

Step 3: Substitute the known values into the formula.
Substituting $a_n = 205$, $a = 7$, and $d = 6$ into the formula:
$205 = 7 + (n - 1)6$

Step 4: Solve for $n$.
Subtract $7$ from both sides of the equation:
$205 - 7 = (n - 1)6$
$198 = (n - 1)6$

Divide both sides by $6$:
$\frac{198}{6} = n - 1$
$33 = n - 1$

Add $1$ to both sides to isolate $n$:
$n = 33 + 1$
$n = 34$

Justification:
Since the number of terms must be a positive integer, and our result $n = 34$ satisfies this condition, the calculation is consistent with the properties of an Arithmetic Progression.

Final Answer: The number of terms in the given AP is 34.

Given: A sequence of numbers $1^2, 3^2, 5^2, 7^2, \dots$

To Find: Determine if the sequence forms an Arithmetic Progression (AP). If it does, find the common difference $d$ and the next three terms.

Step 1: Evaluating the terms of the sequence

Let the sequence be denoted by $a_1, a_2, a_3, a_4, \dots$

Calculating the values of the given terms:

$a_1 = 1^2 = 1$

$a_2 = 3^2 = 9$

$a_3 = 5^2 = 25$

$a_4 = 7^2 = 49$

The sequence is $1, 9, 25, 49, \dots$

Step 2: Checking for a common difference

A sequence is an AP if the difference between consecutive terms is constant, i.e., $a_{n+1} - a_n = d$ for all $n$.

Calculate the difference between the first and second terms ($d_1$):

$d_1 = a_2 - a_1 = 9 - 1 = 8$

Calculate the difference between the second and third terms ($d_2$):

$d_2 = a_3 - a_2 = 25 - 9 = 16$

Calculate the difference between the third and fourth terms ($d_3$):

$d_3 = a_4 - a_3 = 49 - 25 = 24$

Step 3: Conclusion on the nature of the sequence

Since $d_1 \neq d_2 \neq d_3$ (specifically, $8 \neq 16 \neq 24$), the difference between consecutive terms is not constant.

[Definition of an Arithmetic Progression: A sequence is an AP if and only if the difference between any two consecutive terms is constant.]

Final Answer: The given sequence $1^2, 3^2, 5^2, 7^2, \dots$ does not form an Arithmetic Progression because the common difference is not constant.

Given:

Ramkali's savings in the first week ($a_1$) = ₹ $5$.

The weekly increase in savings ($d$) = ₹ $1.75$.

The savings in the $n$th week ($a_n$) = ₹ $20.75$.

To Find:

The value of $n$ (the number of weeks).

Step 1: Identifying the Arithmetic Progression (AP)

Since the savings increase by a constant amount each week, the sequence of savings forms an Arithmetic Progression.

Let the first term be $a = 5$.

Let the common difference be $d = 1.75$.

The $n$th term of an AP is given by the formula: $a_n = a + (n - 1)d$. [Formula for the $n$th term of an AP]

Step 2: Formulating the Equation

Substitute the given values into the formula:

$20.75 = 5 + (n - 1) \times 1.75$

Step 3: Solving for $n$

Subtract $5$ from both sides of the equation:

$20.75 - 5 = (n - 1) \times 1.75$

$15.75 = (n - 1) \times 1.75$

Divide both sides by $1.75$ to isolate the term $(n - 1)$:

$\frac{15.75}{1.75} = n - 1$

To simplify the division, multiply both numerator and denominator by $100$:

$\frac{1575}{175} = n - 1$

Perform the division:

$1575 \div 175 = 9$

$9 = n - 1$

Add $1$ to both sides to solve for $n$:

$n = 9 + 1$

$n = 10$

Step 4: Verification

If $n = 10$, then $a_{10} = 5 + (10 - 1) \times 1.75$

$a_{10} = 5 + 9 \times 1.75$

$a_{10} = 5 + 15.75$

$a_{10} = 20.75$

[Since the calculated $a_{10}$ matches the given $a_n$, the value of $n$ is correct.]

Final Answer: The value of $n$ is $10$.

Given:

The first term of the Arithmetic Progression (AP), denoted by $a = 17$.

The last term of the AP, denoted by $l$ or $a_n = 350$.

The common difference of the AP, denoted by $d = 9$.

To Find:

1. The number of terms in the AP, denoted by $n$.

2. The sum of all terms in the AP, denoted by $S_n$.

Step 1: Finding the number of terms ($n$)

We use the formula for the $n^{th}$ term of an Arithmetic Progression:

$a_n = a + (n - 1)d$

Substituting the given values into the formula:

$350 = 17 + (n - 1)9$

Subtract $17$ from both sides of the equation:

$350 - 17 = (n - 1)9$

$333 = (n - 1)9$

Divide both sides by $9$:

$\frac{333}{9} = n - 1$

$37 = n - 1$

Add $1$ to both sides to solve for $n$:

$n = 37 + 1$

$n = 38$

[Since the number of terms must be a positive integer, $n=38$ is valid.]

Step 2: Finding the sum of the terms ($S_n$)

We use the formula for the sum of the first $n$ terms of an AP when the first and last terms are known:

$S_n = \frac{n}{2}(a + l)$

Substituting the values $n = 38$, $a = 17$, and $l = 350$:

$S_{38} = \frac{38}{2}(17 + 350)$

Simplify the fraction and the expression inside the parentheses:

$S_{38} = 19(367)$

Perform the multiplication:

$19 \times 367 = 6973$

Step 3: Verification of calculation

$19 \times 300 = 5700$

$19 \times 60 = 1140$

$19 \times 7 = 133$

$5700 + 1140 + 133 = 6973$

Final Answer: There are 38 terms in the AP, and their sum is 6973.

Given: The sum of the first $n$ terms of an Arithmetic Progression (AP) is given by the formula $S_n = 4n - n^2$.

To Find:
1. The first term ($a_1$ or $S_1$).
2. The sum of the first two terms ($S_2$).
3. The second term ($a_2$).
4. The third term ($a_3$).
5. The tenth term ($a_{10}$).
6. The $n$th term ($a_n$).

Step 1: Finding the first term ($a_1$)
By definition, the sum of the first term is the first term itself.
Given $S_n = 4n - n^2$.
For $n = 1$:
$S_1 = 4(1) - (1)^2$
$S_1 = 4 - 1 = 3$
Thus, the first term $a_1 = 3$.

Step 2: Finding the sum of the first two terms ($S_2$)
For $n = 2$:
$S_2 = 4(2) - (2)^2$
$S_2 = 8 - 4 = 4$
Thus, the sum of the first two terms is $4$.

Step 3: Finding the second term ($a_2$)
We know that the sum of the first two terms is the sum of the first term and the second term: $S_2 = a_1 + a_2$.
Substituting the known values:
$4 = 3 + a_2$
$a_2 = 4 - 3 = 1$
Thus, the second term $a_2 = 1$.

Step 4: Finding the third term ($a_3$)
First, calculate the sum of the first three terms ($S_3$):
$S_3 = 4(3) - (3)^2 = 12 - 9 = 3$
The $n$th term of an AP can be found using the relation $a_n = S_n - S_{n-1}$.
For $n = 3$:
$a_3 = S_3 - S_2$
$a_3 = 3 - 4 = -1$
Thus, the third term $a_3 = -1$.

Step 5: Finding the tenth term ($a_{10}$)
Using the relation $a_n = S_n - S_{n-1}$:
$a_{10} = S_{10} - S_9$
Calculate $S_{10}$: $S_{10} = 4(10) - (10)^2 = 40 - 100 = -60$
Calculate $S_9$: $S_9 = 4(9) - (9)^2 = 36 - 81 = -45$
$a_{10} = -60 - (-45)$
$a_{10} = -60 + 45 = -15$
Thus, the tenth term $a_{10} = -15$.

Step 6: Finding the $n$th term ($a_n$)
Using the relation $a_n = S_n - S_{n-1}$:
$S_n = 4n - n^2$
$S_{n-1} = 4(n-1) - (n-1)^2$
Expand $S_{n-1}$:
$S_{n-1} = 4n - 4 - (n^2 - 2n + 1)$
$S_{n-1} = 4n - 4 - n^2 + 2n - 1$
$S_{n-1} = -n^2 + 6n - 5$
Now, subtract $S_{n-1}$ from $S_n$:
$a_n = (4n - n^2) - (-n^2 + 6n - 5)$
$a_n = 4n - n^2 + n^2 - 6n + 5$
$a_n = 5 - 2n$

Final Answer:
The first term ($a_1$) is 3.
The sum of the first two terms ($S_2$) is 4.
The second term ($a_2$) is 1.
The third term ($a_3$) is -1.
The tenth term ($a_{10}$) is -15.
The $n$th term ($a_n$) is $5 - 2n$.

Given: An Arithmetic Progression (AP) where the 17th term ($a_{17}$) exceeds the 10th term ($a_{10}$) by 7.

To find: The common difference ($d$) of the Arithmetic Progression.

Step 1: Defining the general term of an Arithmetic Progression

The $n^{th}$ term of an Arithmetic Progression is given by the formula:
$a_n = a + (n - 1)d$
where:
$a$ = the first term of the AP
$d$ = the common difference
$n$ = the position of the term in the sequence

Step 2: Expressing the 17th and 10th terms using the formula

For the 17th term ($n = 17$):
$a_{17} = a + (17 - 1)d$
$a_{17} = a + 16d$ --- (Equation 1)

For the 10th term ($n = 10$):
$a_{10} = a + (10 - 1)d$
$a_{10} = a + 9d$ --- (Equation 2)

Step 3: Formulating the equation based on the given condition

The problem states that the 17th term exceeds the 10th term by 7. Mathematically, this is expressed as:
$a_{17} - a_{10} = 7$

Step 4: Substituting the expressions into the equation

Substitute Equation 1 and Equation 2 into the condition established in Step 3:
$(a + 16d) - (a + 9d) = 7$

Step 5: Solving for the common difference ($d$)

Distribute the negative sign across the terms in the second parenthesis:
$a + 16d - a - 9d = 7$

Group the like terms:
$(a - a) + (16d - 9d) = 7$

Simplify the expression:
$0 + 7d = 7$
$7d = 7$

Divide both sides by 7 to isolate $d$:
$d = \frac{7}{7}$
$d = 1$

Final Answer: The common difference of the Arithmetic Progression is 1.

Given: A sequence of numbers: $0, -4, -8, -12, \dots$

To find:
1. Determine if the given sequence forms an Arithmetic Progression (AP).
2. If it forms an AP, find the common difference ($d$).
3. Write the next three terms of the sequence.

Step 1: Definition of an Arithmetic Progression
A sequence $a_1, a_2, a_3, \dots, a_n$ is said to be an Arithmetic Progression if the difference between consecutive terms remains constant. This constant difference is called the common difference ($d$), defined as:
$d = a_{n} - a_{n-1}$ for all $n > 1$.

Step 2: Calculating differences between consecutive terms
Let the given terms be:
$a_1 = 0$
$a_2 = -4$
$a_3 = -8$
$a_4 = -12$

Calculate the differences:
Difference 1 ($d_1$): $a_2 - a_1 = -4 - 0 = -4$
Difference 2 ($d_2$): $a_3 - a_2 = -8 - (-4) = -8 + 4 = -4$
Difference 3 ($d_3$): $a_4 - a_3 = -12 - (-8) = -12 + 8 = -4$

Step 3: Verification of AP
[Since $d_1 = d_2 = d_3 = -4$, the difference between consecutive terms is constant.]
Therefore, the given sequence forms an Arithmetic Progression with a common difference $d = -4$.

Step 4: Finding the next three terms
To find the next terms, we add the common difference ($d = -4$) to the last known term ($a_4 = -12$).
Let the next three terms be $a_5, a_6,$ and $a_7$.

Calculation for $a_5$:
$a_5 = a_4 + d = -12 + (-4) = -16$

Calculation for $a_6$:
$a_6 = a_5 + d = -16 + (-4) = -20$

Calculation for $a_7$:
$a_7 = a_6 + d = -20 + (-4) = -24$

Final Answer:
The sequence forms an AP.
The common difference $d = -4$.
The next three terms are -16, -20, -24.

Given: An Arithmetic Progression (AP) sequence: $18, 15\frac{1}{2}, 13, \dots, -47$.

To find: The number of terms ($n$) in the given AP.

Step 1: Identify the parameters of the Arithmetic Progression.

An Arithmetic Progression is defined by its first term ($a$) and its common difference ($d$).

The first term is $a = 18$.

The second term is $a_2 = 15\frac{1}{2} = \frac{31}{2}$.

The common difference ($d$) is calculated as $d = a_2 - a_1$:

$d = \frac{31}{2} - 18$

$d = \frac{31}{2} - \frac{36}{2}$ [Converting 18 to a fraction with denominator 2]

$d = -\frac{5}{2}$

Step 2: State the General Term Formula.

The $n^{th}$ term of an AP is given by the formula:

$a_n = a + (n - 1)d$

Where:

• $a_n$ is the last term (given as $-47$)
• $a$ is the first term ($18$)
• $n$ is the number of terms
• $d$ is the common difference ($-\frac{5}{2}$)

Step 3: Substitute the values into the formula and solve for $n$.

$-47 = 18 + (n - 1)\left(-\frac{5}{2}\right)$

Subtract 18 from both sides:

$-47 - 18 = (n - 1)\left(-\frac{5}{2}\right)$

$-65 = (n - 1)\left(-\frac{5}{2}\right)$

Multiply both sides by $-\frac{2}{5}$ to isolate $(n - 1)$:

$-65 \times \left(-\frac{2}{5}\right) = n - 1$

$\frac{130}{5} = n - 1$ [Since $65 \div 5 = 13$]

$26 = n - 1$

Add 1 to both sides:

$n = 26 + 1$

$n = 27$

Step 4: Conclusion.

Since $n$ represents the count of terms, it must be a positive integer. Our result $n = 27$ satisfies this condition.

Final Answer: The number of terms in the given AP is 27.

Given:

The first term of the Arithmetic Progression (AP), denoted by $a = 5$.

The last term of the AP, denoted by $l$ or $a_n = 45$.

The sum of the $n$ terms of the AP, denoted by $S_n = 400$.

To Find:

1. The number of terms ($n$).

2. The common difference ($d$).

Step 1: Finding the number of terms ($n$)

We use the formula for the sum of an AP when the first and last terms are known:

$S_n = \frac{n}{2}(a + l)$

[Substituting the given values into the formula]

$400 = \frac{n}{2}(5 + 45)$

$400 = \frac{n}{2}(50)$

[Simplifying the expression inside the parentheses and dividing 50 by 2]

$400 = n \times 25$

$n = \frac{400}{25}$

$n = 16$

[Since $n$ represents the number of terms, it must be a positive integer]

Step 2: Finding the common difference ($d$)

We use the formula for the $n^{th}$ term of an AP:

$a_n = a + (n - 1)d$

[Substituting the known values: $a_n = 45$, $a = 5$, and $n = 16$]

$45 = 5 + (16 - 1)d$

$45 = 5 + 15d$

[Subtracting 5 from both sides of the equation]

$45 - 5 = 15d$

$40 = 15d$

[Dividing both sides by 15 to solve for $d$]

$d = \frac{40}{15}$

[Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, 5]

$d = \frac{8}{3}$

Final Answer: The number of terms is 16 and the common difference is $\frac{8}{3}$.