Class 10 Maths Chapter 9
Arithmetic Progression
Important Questions

Class 10 Mathematics chapter 9, ‘Arithmetic Progression’ is a crucial topic that necessitates a comprehensive grasp and ample practice. To assist students in their exam preparation, we have assembled a collection of essential questions pertaining to Class 10 Arithmetic Progressions.
An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed value (common difference, 'd') to the previous term. It's a fundamental concept in mathematics. Key formulas include finding the nth term (an = a + (n - 1) * d) and the sum of the first 'n' terms (Sn = n/2 * [2a + (n - 1) * d]). APs are useful in solving various mathematical and real-world problems, helping to understand patterns and relationships between numbers. Remember to carefully read the problem of arithmetic progression class 10 ICSE questions apply the relevant formula, and double-check your answers for accuracy when working with APs.

Introduction

An arithmetic progression (AP), in Arithmetic Progression class 10 ICSE Board is also known as an arithmetic sequence, is a fundamental concept in mathematics that you'll encounter in ICSE Class 10 mathematics. It's a sequence of numbers in which each term after the first is obtained by adding a fixed constant value to the previous term. This constant value is called the "common difference."
The general form of an arithmetic progression is represented as:
, a + d, a + 2d, a + 3d, ...
Where:'a' is the first term in the sequence.
'd' is the common difference between consecutive terms.
Key points to understand about arithmetic progressions:
Common Difference (d): The difference between any two consecutive terms in an AP is always the same. It can be positive, negative, or zero.
Formula for nth Term (an): The nth term of an arithmetic progression can be calculated using the formula:
an = a + (n - 1) * d
Sum of First 'n' Terms (Sn): You can find the sum of the first 'n' terms of an AP using the formula:
Sn = (n/2) * [2a + (n - 1) * d]
nth Term from the End: To find the nth term from the end of an AP, you can use the formula:an' = a + (N - n) * d
Where 'an'' is the nth term from the end, 'N' is the total number of terms, 'n' is the position from the end, 'a' is the first term, and 'd' is the common difference.
Finding 'd' If you know the first term (a), the nth term (an), and the total number of terms (n), you can find the common difference 'd' using the formula:d = (an - a) / (n - 1)
Finding 'n': If you know the first term (a), the common difference (d), and a term (an) in the sequence, you can find the position 'n' using the formula:
n = (an - a) / d + 1

What is Arithmetic Progression? 

An arithmetic progression (AP), Arithmetic Progression class 10 ICSE Board also known as an arithmetic sequence, is a fundamental concept in ICSE Class 10 mathematics. It's a sequence of numbers in which each term after the first is obtained by adding a fixed constant value to the previous term. This constant value is called the "common difference."
The general form of an arithmetic progression is represented as:
a, a + d, a + 2d, a + 3d, ...
Where:
'a' is the first term in the sequence.
'd' is the common difference between consecutive terms.
Key points to understand about arithmetic progressions:
Common Difference (d): The difference between any two consecutive terms in an AP is always the same. It can be positive, negative, or zero.
Formula for nth Term (an): The nth term of an arithmetic progression can be calculated using the formula:
an = a + (n - 1) * d
Sum of First 'n' Terms (Sn): You can find the sum of the first 'n' terms of an AP using the formula:
Sn = (n/2) * [2a + (n - 1) * d]
nth Term from the End: To find the nth term from the end of an AP, you can use the formula:
an' = a + (N - n) * d
Where 'an'' is the nth term from the end, 'N' is the total number of terms, 'n' is the position from the end, 'a' is the first term, and 'd' is the common difference.
Finding 'd': If you know the first term (a), the nth term (an), and the total number of terms (n), you can find the common difference 'd' using the formula:
d = (an - a) / (n - 1)
Finding 'n': If you know the first term (a), the common difference (d), and a term (an) in the sequence, you can find the position 'n' using the formula:
n = (an - a) / d + 1

Class 10 Arithmetic Progression Important Questions and Answers

Q1. Which term of the A.P. 1, 4, 7, 10, .... is 58?

Options

(a) 18
(b) 19
(c) 20
(d) 21

Ans. (c) 20

Explanation:
A.P. = 1,4,7,10………
Here, first term (a) = 1
And common difference (d) = 4 - 1 = 3
Let n^th  term of the given A.P. is 58.
⇒ 58 = a + (n-1)d
⇒ 58 = 1 + (n-1)× 3 
⇒ n - 1 = \(\frac{57}{3}\) = 19
⇒ n = 20
So, 58 is 20^th term of the A.P.

Q2. The sum of 25 terms of an A.P. - \(\dfrac{2}{3}\), - \(\dfrac{2}{3}\), - \(\dfrac{2}{3}\) is :

Options

(a) 0
(b) -\(\frac{2}{3}\)
(c) -\(\frac{50}{3}\)
(d) -50

Ans. (c)

Explanation:
sum of 25 terms of an A.P. \(\dfrac{-2}{3}, \dfrac{-2}{3},\dfrac{-2}{3}\) is
= \(\dfrac{n}{2}\) [2a+(n-1)d]
=\(\dfrac{25}{2}\begin{bmatrix} 2 × \left(\dfrac{-2}{3}\right) + (25 - 1 ) × 0\end{bmatrix}\)
= \(\dfrac{}{}\begin{bmatrix} -\dfrac{4}{3}\end{bmatrix}\) = \(\dfrac{-50}{3}\)

Q3. In an Arithmetic Progression (A.P.) the fourth and sixth terms are 8 and 14 respectively. Find the :
(a) First term
(b) Common difference
(c) Sum of the first 20 terms.

Explanation:
 Let, a and d be the first term and common difference of the given A.P.
Then, a⁴ = 8 and a⁶ = 14 (Given)
⇒ a + 3d = 8 ...(i)
and a + 5d = 14 ...(ii)
On subtracting equation (i) from (ii), we get
2d = 6
⇒ d = 3
On putting d = 3 in equation (i), we get
a + 3 × 3 = 8
⇒ a = 8 – 9 = – 1
(a) First term (a) = – 1. Ans.
(b) Common difference (d) = 3. Ans.(c) Sum of first 20 terms (S20)
∵ S_n = \(\dfrac{n}{2}\) [2a+(n–1)d]
∴ S₂₀ = \(\dfrac{20}{2}\) [2× (-1) +(20–1)×3]
= 10(-2 + 57 )
= 550.

Q4.  Determine the 10th term from the last term (towards the first term) of the A.P. 10, 7, 4, … , – 62.

Explanation:
Here a = 10, d = 7 – 10 = – 3
and last term l = – 62
10th term from the last term
i.e., n = 10
Required term = l – (n – 1) d
= – 62 – (10 – 1) (– 3)
= – 62 + 27
= – 35
Therefore, the 10th term from the last term is – 35.

Q5. The 10th term of an A.P. is – 15 and 31st term is – 57, find the 15th term.

Explanation:
Let a be the first term and d be the common difference of the A.P. Then from the formula :
t_n = a + (n–1)d,
we have         t₁₀ ​ =a + (10–1)d = a + 9d
         t₃₁ ​ = a + (31–1)d = a + 30d
we have,          a + 9d = – 15         …(i)
         a + 30d = –57         …(ii)
Solve equations (i) and (ii) to get the values of a and d.
Subtracting (i) from (ii), we have
         21d = –57 + 15 = –42
∴         d = \(\dfrac{-42}{21}\) = -2
Again from (i), a = – 15 – 9d
         = –15 – 9(–2)
         = –15 + 18 = 3
Now, t₁₅ = a + (15 – 1) d
         = 3 + 14(–2) = –25

ICSE Class 10 Maths Chapter wise Important Questions

Conclusion

In ICSE Class 10 Mathematics, arithmetic progressions (APs) are fundamental. AP is a sequence with a constant difference between consecutive terms (d). Key formulas include:The nth term: a_n= a + (n - 1) * d
Sum of first 'n' terms: S_n = \(\frac{n}{2}\)[2a₁ + (n - 1)d].
Sum with a specific number of terms: S = \(\frac{n}{2}\)[2a₁ + (n - 1)d]. If you need to practise more arithmetic progression class 10 ICSE questions, then oswal.io gives you a wide range of questions to practise.

Frequently Asked Questions

Q1 : What is an Arithmetic Progression (AP)?

Ans: An AP is a sequence of numbers where the difference between any two consecutive terms is constant.

Q2: What is the common difference in an AP?

Ans:  The common difference is the fixed value by which consecutive terms in an AP increase or decrease.

Q3 : How can I find the nth term of an AP?

Ans: Use the formula: a_n= a₁+ ( n-1 )d, where  an is the nth term,  a1 is the first term, and d is the common difference.

Q4 : What is the sum of the first 'n' terms of an AP?

Ans: You can calculate it with the formula: S_n = \(\dfrac{n}{2}\)[2a₁ + (n - 1)d].

Q5 : How do I find the number of terms in an AP given the first and last terms?

Ans: Use the formula: n = \(\dfrac{a-a}{d}\) + 1