Practice Questions

Apply the sum formula to find the sum of an AP with 15 terms, whose first term is 10 and last term is 80.

Recall the formula for the $n$th term ($a_n$) of an AP.

Name the general term of an AP.

Consider the sequence $1^2, 2^2, 3^2, 4^2, \ldots$, which is $1, 4, 9, 16, \ldots$. This is not an AP. Propose a modification to the third term of this sequence so that the first three terms, $1, 4, \ldots$, form an AP. Justify your proposed value.

Define an Arithmetic Progression (AP).

A sequence is defined by $a_n = 5$ for all $n \ge 1$. Justify why this sequence is an Arithmetic Progression.

Recall the two formulas for the sum of the first $n$ terms of an AP.

Apply the concept of an AP to find the common difference if the 3rd term is 12 and the 5th term is 20.

Calculate the 21st term of the Arithmetic Progression whose first term is $-5$ and the common difference is $2.5$.

To find the number of terms in the AP $7, 13, 19, \ldots, 205$, a student writes the equation $205 = 7 + (n) \times 6$. Critique this approach. Identify the error and provide the correct formulation to solve the problem.

Identify the first term $a$ and the common difference $d$ for the AP: $3, 1, -1, -3, \ldots$

A list of numbers is given by $a, a^2, a^3, \ldots$ for $a \neq 0$ and $a \neq 1$. Justify why this is not an Arithmetic Progression.

A spiral is made of successive semicircles, with centers alternately at A and B. The radii of the semicircles form an AP: $0.5 \text{ cm}, 1.0 \text{ cm}, 1.5 \text{ cm}, \ldots$. Formulate an expression for the total length of a spiral made up of 13 consecutive semicircles and evaluate this length. (Use $\pi = \frac{22}{7}$).

Two APs have the same common difference. The first term of one is 3 and that of the other is 8. Evaluate the difference between their 10th terms and their 100th terms. Based on your findings, formulate a general statement about the difference between the $n$-th terms of two such APs.

Evaluate the statement: "If the $p$-th term of an AP is $q$ and the $q$-th term is $p$, then its common difference is always $-1$." Justify your answer with a proof.

A manufacturer of mobile phones produced 800 units in the fourth year and 1130 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, calculate (i) the production in the first year, and (ii) the total production in the first 10 years.

Explain how to determine if a given list of numbers forms an AP.

Describe the general form of an AP and explain what each component represents.

Using the formula $a_n = a + (n-1)d$, find the 20th term of the AP: $2, 7, 12, 17, \ldots$

Solve for the number of terms in the AP: $7, 13, 19, \ldots, 205$.

Identify the common difference for the AP: $-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \ldots$

List the first four terms of the AP when the first term $a = 10$ and the common difference $d = -3$.

Summarize the key properties of the common difference ($d$) in an AP.

Explain the difference between a finite AP and an infinite AP, providing one example for each.

Examine if the sequence $\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots$ forms an Arithmetic Progression.

In an auditorium, the seats are arranged in rows. The first row has 20 seats, the second row has 24 seats, the third row has 28 seats, and so on. If there are 30 rows in the auditorium, solve for the total number of seats.

Create an Arithmetic Progression with a positive first term and a negative common difference, such that its 8th term is the first negative term. State the first term and common difference you chose and justify your choice.

Compare the two APs: $3, 7, 11, 15, \ldots$ and $63, 65, 67, 69, \ldots$. Solve for the value of $n$ for which their $n$th terms are equal.

Apply your knowledge of APs to calculate the 10th term from the end of the AP: $4, 9, 14, \ldots, 254$.

List the first five terms of an AP whose first term is $a = -1.25$ and common difference is $d = -0.25$.

Calculate the sum of the first 20 terms of a sequence whose $n$th term is given by the formula $a_n = 5 - 2n$.

Solve for the first term and the common difference of an AP where the 4th term is 11 and the sum of the 5th and 7th terms is 34.

A student claims that for the AP $5, 12, 19, \ldots$, the term $250$ is a part of the sequence. Evaluate this claim by determining if the term number $n$ is a positive integer. Justify your conclusion.

Two students, Rohan and Priya, are asked to find the sum of the AP: $2, 5, 8, \ldots$ up to 10 terms. Rohan calculates the 10th term first, $a_{10} = 2 + (10-1)3 = 29$, and then finds the sum as $S_{10} = \frac{10}{2}(2+29) = 155$. Priya directly calculates the sum as $S_{10} = \frac{10}{2}[2(2)+(10-1)3] = 155$. Critique both methods. Justify which method is more universally applicable and explain the potential pitfall of Rohan's method.

Explain why the sequence $a, a^2, a^3, a^4, \ldots$ (for $a \neq 0, 1$) is not an AP.

The sum of the first $n$ terms of a sequence is given by $S_n = 3n^2 + 5n$. Analyze this sequence to determine if it is an AP. If it is, calculate its common difference and its 10th term.

The sum of the first $n$ terms of a sequence is given by the formula $S_n = 3n^2 - n$. Justify whether this sequence is an Arithmetic Progression. If it is, formulate the expression for its common difference.

Calculate the sum of all three-digit numbers which are divisible by 9.

The numbers 2 and 34 are the first and last terms of an AP. Propose a method to insert five numbers between 2 and 34 such that the entire sequence forms an AP. Find these five numbers.

Summarize the minimum information required to fully describe a unique Arithmetic Progression. Explain why knowing only one piece of information is insufficient.

Analyze the following situation: A person repays a loan of ₹3250 by paying ₹20 in the first month and then increases the payment by ₹15 every month. Does the monthly payment form an AP? If so, calculate how many months it will take to clear the loan.

Design a problem involving the angles of a quadrilateral where the angles are in an Arithmetic Progression. The problem should require the user to find all four angles, given a relationship that the largest angle is three times the smallest angle. Formulate and solve the problem you have designed.

The sum of three consecutive terms in an AP is 21. The product of the first and the third term is 45. Formulate a pair of equations representing this situation, assuming the terms are $a-d, a, a+d$. Solve these equations to find the three terms.

Design a real-world savings plan problem for a student where the monthly savings form an Arithmetic Progression. Formulate a question based on your plan that requires calculating the total savings after 2 years (24 months), and then solve it. The initial monthly saving and the monthly increment must be realistic values.

Solve for the number of terms of the AP $63, 60, 57, \ldots$ that must be taken so that their sum is 693.