What is the nth term of an Arithmetic Progression?

The nth term of an AP is given by an = a + (n−1)d, where a is the first term and d is the common difference.

How to find the sum of n terms of an AP?

The sum of n terms (Sn) is given by Sn = n/2 × [2a + (n−1)d] or Sn = n/2 × (a + l), where l is the last term.

Are these AP notes useful for SSC, NEET, and Police Bharti exams?

Yes, these Class 10 Arithmetic Progression notes are useful for CBSE students as well as SSC, NEET, Railway, and Police Bharti aspirants.

Arithmetic Progressions — Class 10 Maths Notes

1. What is an Arithmetic Progression (AP)?

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This constant is called the common difference (d).

General form: a, a + d, a + 2d, a + 3d, ... where a = first term and d = common difference.

2. n-th Term of an AP

The n-th term (T_n) of an AP is given by:

T_n = a + (n − 1)d

Where a = first term, d = common difference. Use this to find any term in the sequence.

3. Sum of First n Terms (S_n)

Sum of first n terms of an AP can be calculated using two equivalent formulas:

S_n = (n/2) [2a + (n − 1)d] = (n/2)(a + l)

Where l = n-th term (last term). Both formulas are useful; pick the one that fits the known values.

Derivation (short): Add S_n forwards and backwards and pair terms to obtain n pairs each summing to (a + l).

4. Useful Properties & Tips

If terms a_m and a_n are known, common difference d = (a_n − a_m)/(n − m).
Arithmetic mean of two numbers = (their sum)/2; numbers equally spaced form an AP.
If three numbers are in AP, the middle one is the arithmetic mean of the other two.
For negative d, sequence decreases; for positive d, increases.

5. Solved Examples

Example 1. Find the 20th term of the AP: 3, 7, 11, ...

a = 3, d = 4 ⇒ T₂₀ = 3 + (20 − 1)×4 = 3 + 76 = 79.

Example 2. Sum of first 15 terms of AP 2, 5, 8, ...

a = 2, d = 3, n = 15 ⇒ S₁₅ = (15/2)[2×2 + (15−1)×3] = (15/2)[4 + 42] = (15/2)×46 = 15×23 = 345.

Example 3. If the 5th term of an AP is 12 and the 12th term is 33, find a and d.

Use T_n = a + (n−1)d: a + 4d = 12 and a + 11d = 33 ⇒ subtract ⇒ 7d = 21 ⇒ d = 3. Then a = 12 − 4×3 = 0.

6. Practice Questions

Find the 50th term of the AP: 1, 4, 7, ...
Sum of first 20 terms of AP: 5, 9, 13, ...
Three numbers are in AP and their sum is 27. If the common difference is 3, find the numbers.
If T₈ = 26 and T₁₂ = 38 in an AP, find a and d.

7. Exam Tips

Always identify a and d from the first two terms (unless given otherwise).
When dealing with sums, prefer S_n = (n/2)(a + l) if last term is known — fewer calculations.
For large n, use formulas directly; avoid term-by-term addition.