Class 10 Maths · Original EZT practice
Arithmetic Progressions (Class 10 Chapter 5): summary and practice
Arithmetic Progressions studies sequences that grow by a fixed step: the common difference d, the nth term a_n = a + (n - 1)d, and the two sum formulas S_n = n/2 (2a + (n - 1)d) and S_n = n/2 (a + l). All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Introduction
Salary increments, savings deposits, seats in a theatre row - many everyday patterns add the same amount each step. That fixed step is what makes a sequence arithmetic.
2. Arithmetic progressions
A sequence is an AP when each term minus the previous one gives the same number, the common difference d. The test works for decreasing sequences too - d just comes out negative.
3. The nth term
Starting from a and adding d a total of n - 1 times gives a_n = a + (n - 1)d. The same formula answers the reverse question: to check whether a number is a term, solve for n and see whether it comes out a whole number.
4. Sum of the first n terms
Pairing the first and last terms gives S_n = n/2 (a + l), and substituting l = a + (n - 1)d gives S_n = n/2 (2a + (n - 1)d). Use the first when you know the last term, the second when you do not.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Find the nth term
Choose the method: a_n = a + (n - 1)d - the first term plus n - 1 steps.
EZT practice question
For the AP 2, 5, 8, ... find the 10th term.
Show the worked solution
- a = 2, d = 5 - 2 = 3.
- a_10 = 2 + (10 - 1) x 3.
- = 2 + 27.
Answer: a_10 = 29.
Check: List a few: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 - the tenth entry matches.
Watch for this: The multiplier is n - 1, not n: the first term is already there before any step.
Now try it yourself
8th term of 7, 11, 15, ...?
Check your answer
35.
7 + 7 x 4 = 35.
Check whether a number is a term
Choose the method: Solve a + (n - 1)d = N for n; N is a term only if n is a whole number.
EZT practice question
Is 50 a term of the AP 2, 5, 8, ...?
Show the worked solution
- Set 2 + (n - 1) x 3 = 50.
- (n - 1) x 3 = 48.
- n - 1 = 16, so n = 17.
Answer: Yes - 50 is the 17th term.
Check: 2 + 16 x 3 = 50 confirms it directly.
Watch for this: If n comes out fractional the number is NOT a term - do not round it.
Now try it yourself
Is 100 a term of 2, 5, 8, ...?
Check your answer
No.
n - 1 = 98/3 is not a whole number.
Sum the first n terms
Choose the method: S_n = n/2 (2a + (n - 1)d).
EZT practice question
Sum of the first 10 terms of 2, 5, 8, ...?
Show the worked solution
- a = 2, d = 3, n = 10.
- S_10 = 10/2 x (2 x 2 + 9 x 3).
- = 5 x (4 + 27) = 5 x 31.
Answer: S_10 = 155.
Check: With a_10 = 29 the other formula agrees: 10/2 x (2 + 29) = 5 x 31 = 155.
Watch for this: Do not drop the 2 in 2a - (a + (n - 1)d) alone is just the last term.
Now try it yourself
S_8 of 7, 11, 15, ...?
Check your answer
168.
8/2 x (14 + 28) = 4 x 42.
Sum using the last term
Choose the method: S_n = n/2 (a + l) when the last term l is known.
EZT practice question
Sum the AP 1, 5, 9, ... up to its 20th term.
Show the worked solution
- a = 1, d = 4.
- Last term: l = 1 + 19 x 4 = 77.
- S_20 = 20/2 x (1 + 77).
Answer: S_20 = 780.
Check: The other formula gives 10 x (2 + 76) = 780 - same answer, as it must.
Watch for this: l means the LAST of the n terms being summed, not any later term.
Now try it yourself
Sum 3, 7, 11, ... to 10 terms.
Check your answer
210.
l = 3 + 36 = 39, so 5 x (3 + 39).
Read a decreasing AP
Choose the method: d is term minus PREVIOUS term - decreasing sequences give negative d.
EZT practice question
Find d and the 6th term of 20, 17, 14, ...
Show the worked solution
- d = 17 - 20 = -3.
- a_6 = 20 + 5 x (-3).
- = 20 - 15.
Answer: d = -3, a_6 = 5.
Check: Listing 20, 17, 14, 11, 8, 5 lands on 5 - matches.
Watch for this: Writing d = 3 (positive) flips every later term upward.
Now try it yourself
5th term of 10, 7, 4, ...?
Check your answer
-2.
10 + 4 x (-3).
Test whether a sequence is an AP
Choose the method: Check that every consecutive difference is the same number.
EZT practice question
Is 3, 7, 12, 18, ... an AP?
Show the worked solution
- Differences: 7 - 3 = 4.
- 12 - 7 = 5.
- 18 - 12 = 6.
Answer: No - the differences 4, 5, 6 are not constant.
Check: An AP needs ONE common difference for the whole sequence; three different differences settle it.
Watch for this: Two equal differences are not enough - check every consecutive pair given.
Now try it yourself
Is 5, 9, 13, 17 an AP?
Check your answer
Yes, d = 4.
Every difference equals 4.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Common difference?
Show answer and reason
d = a_n - a_(n-1).
Same step everywhere makes it arithmetic.
2. nth term?
Show answer and reason
a_n = a + (n - 1)d.
First term plus n - 1 steps.
3. Sum, d known?
Show answer and reason
S_n = n/2 (2a + (n - 1)d).
n pairs averaging the first and last.
4. Sum, last term known?
Show answer and reason
S_n = n/2 (a + l).
Average term times number of terms.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Find the 8th term of 7, 11, 15, ...
- Is 100 a term of 2, 5, 8, ...? Say why in one line.
- Sum the first 8 terms of 7, 11, 15, ...
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Questions about this practice page
- How do I tell whether a sequence is an arithmetic progression?
- Subtract each term from the next. If every difference is the same number, that number is the common difference d and the sequence is an AP; if even one difference changes, it is not.
- Why does the nth term formula use n - 1?
- Because the first term is already in place before any step is taken. Reaching the nth term needs only n - 1 additions of d, so a_n = a + (n - 1)d.
- Which sum formula should I use?
- S_n = n/2 (a + l) when you know the last term of the stretch being summed; S_n = n/2 (2a + (n - 1)d) when you only know a, d and n. Both give the same answer - the first just saves a step.
- Can an AP decrease?
- Yes - then d is negative. The sequence 20, 17, 14, ... has d = -3, and the same formulas work unchanged as long as you keep the sign.
- Are these the NCERT exercise questions for Arithmetic Progressions (Class 10)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.