Class 10 Maths · 2026–27 study draft

Arithmetic Progressions: Find the Pattern, Term and Sum

An arithmetic progression, or AP, is a sequence whose consecutive terms differ by a fixed number. The difference can be positive, negative or zero. APs help model regular savings, rows of seats and quantities that change by a constant amount.

What you will learn

  • Decide whether a sequence has a constant difference.
  • Find a term or its position.
  • Derive and apply the sum formula.
  • Model a daily-life situation and check that term counts are positive integers.

The CBSE 2026–27 syllabus includes motivation for APs, derivation of the nth term and the sum of the first n terms, and their applications to daily life. These notes explain why the formulas work as well as how to use them.

Core concepts

Check the difference, not the ratio

Subtract each term from the next. The sequence 11,7,3,−111,7,3,-1 has common difference −4-4 and is an AP. The sequence 2,4,8,162,4,8,16 is not an AP because its differences are not constant. In a finite list, check every available consecutive difference.

The first term uses zero steps

Starting at first term aa, the second term is a+da+d and the third is a+2da+2d. The nth term follows after n−1n-1 steps, so an=a+(n−1)da_n=a+(n-1)d. A solution for a position nn must be a positive integer; a fractional result means the proposed number is not a term.

Pairing explains the sum

Write the sum forwards and backwards. Each paired column adds to a+la+l, where ll is the last term. There are nn columns, so 2Sn=n(a+l)2S_n=n(a+l). Substitute l=a+(n−1)dl=a+(n-1)d to obtain the version using the common difference.

Important formulas

d=an+1−and=a_{n+1}-a_n
an=a+(n−1)da_n=a+(n-1)d
Sn=n2(a+l)=n2[2a+(n−1)d]S_n=\frac n2(a+l)=\frac n2\left[2a+(n-1)d\right]

Three worked examples

Example 1

Find the 18th term of 6,10,14,18,…6,10,14,18,\ldots.

  1. Identify a=6a=6 and d=10−6=4d=10-6=4.
  2. There are 18−1=1718-1=17 increases from the first term to the eighteenth.
  3. Compute a18=6+17(4)=74a_{18}=6+17(4)=74.

Answer: The 18th term is 7474.

Example 2

Which term of 41,36,31,26,…41,36,31,26,\ldots equals −24-24?

  1. Here a=41a=41 and d=−5d=-5. Set −24=41+(n−1)(−5)-24=41+(n-1)(-5).
  2. Rearrange: −65=−5(n−1)-65=-5(n-1), giving n−1=13n-1=13 and n=14n=14.
  3. Check: 41+13(−5)=−2441+13(-5)=-24. The position is a positive integer.

Answer: −24-24 is the 14th term.

Example 3

A student saves ₹30 in the first week and ₹10 more each following week. How much is saved over 12 weeks?

  1. The weekly amounts form an AP with a=30a=30, d=10d=10 and n=12n=12.
  2. The twelfth week's amount is 30+11(10)=14030+11(10)=140 rupees.
  3. Add all weekly amounts: S12=122(30+140)=1020S_{12}=\frac{12}{2}(30+140)=1020 rupees. This is the total, not the last week's saving.

Answer: The student saves ₹1,020 in total.

Common errors to avoid

  • Use n−1n-1, not nn, when calculating the nth term.
  • A decreasing AP has a negative common difference; keep its sign.
  • Distinguish the amount in week n from the total over n weeks.
  • Never round a non-integer position and claim the number belongs to the AP.

Try it yourself

Attempt these original questions before checking the answer and reasoning.

  1. Find the 20th term of 3,8,13,18,…3,8,13,18,\ldots.

    Check answer and reasoning

    9898.

    a20=3+19(5)=98a_{20}=3+19(5)=98.

  2. Find the sum of the first 15 terms of 2,5,8,11,…2,5,8,11,\ldots.

    Check answer and reasoning

    345345.

    a15=44a_{15}=44 and S15=15(2+44)/2=345S_{15}=15(2+44)/2=345.

  3. Is 100100 a term of 7,13,19,25,…7,13,19,25,\ldots?

    Check answer and reasoning

    No.

    100=7+6(n−1)100=7+6(n-1) gives n=16.5n=16.5, not a positive integer.

Related chapter guides

Quadratic Equations · Introduction to Trigonometry

Quadratic equations help when an AP sum creates an equation in the number of terms. Trigonometry offers a different application of careful substitution.

Explore all Class 10 Maths chapters

Official syllabus and draft review

Aligned to the official CBSE Class X Mathematics syllabus for 2026–27 (041 and 241). Check the CBSE curriculum portal for updates and your school's guidance.

Original Mathslume study drafts, checked against the syllabus with independently calculated answers. Not official CBSE/NCERT solutions or an endorsement. A qualified teacher's editorial sign-off is still recommended before publication.

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