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If the common difference of an AP is 5, then what is the value of a₁₈ - a₁₃?

25. a₁₈ - a₁₃ = [a + 17d] - [a + 12d] = 5d = 5 × 5 = 25.

Mathematics · Class 9-10 · CBSE

Arithmetic Progressions: Formulas, Derivations and Word Problems

Learn arithmetic progressions (AP): common difference d, derivation of the nth term formula an = a + (n-1)d, Gauss derivation of sum of n terms Sn = (n/2)[2a + (n-1)d], and practical word problems.

1Definition and Components of an AP

AnArithmetic Progression (AP)is a sequence of numbers in which each term is obtained by adding a fixed constant number, called theCommon Difference (d), to the preceding term, except the first term (a).

Common Difference Property

d = a_{n} - a_{n-1} = a_2 - a_1 = a_3 - a_2

Types of Common Difference (d)

Value of d	Sequence Trend	Example AP Series	Significance
Positive (d > 0)	Strictly increasing sequence	$3, 7, 11, 15, 19... (d = +4)$	Terms grow progressively larger
Negative (d < 0)	Strictly decreasing sequence	$100, 90, 80, 70... (d = -10)$	Terms decrease towards negative values
$\text{Zero}\ (d = 0)$	Constant stationary sequence	$5, 5, 5, 5, 5... (d = 0)$	Every term is identical to the first term

Positive (d > 0)
Sequence Trend
Strictly increasing sequence
Example AP Series
$3, 7, 11, 15, 19... (d = +4)$
Significance
Terms grow progressively larger
Negative (d < 0)
Sequence Trend
Strictly decreasing sequence
Example AP Series
$100, 90, 80, 70... (d = -10)$
Significance
Terms decrease towards negative values
$\text{Zero}\ (d = 0)$
Sequence Trend
Constant stationary sequence
Example AP Series
$5, 5, 5, 5, 5... (d = 0)$
Significance
Every term is identical to the first term

2The nth Term Formula and Selection of Terms

General nth Term of an AP

a_n = a + (n - 1) \cdot d

$a_n$ = The nth term (or last term l)
a = The first term
n = Position index (must be a positive integer 1, 2, 3...)
d = Common difference

Clever Selection of Terms in Problem Solving

Number of Terms	Optimal Symmetric Form	Common Difference	Sum Simplification Benefit
3 Terms	a - d, a, a + d	d	Sum = 3a (d cancels instantly)
4 Terms	a - 3d, a - d, a + d, a + 3d	2d	Sum = 4a (d cancels instantly)
5 Terms	a - 2d, a - d, a, a + d, a + 2d	d	Sum = 5a (d cancels instantly)

3 Terms
Optimal Symmetric Form
a - d, a, a + d
Common Difference
d
Sum Simplification Benefit
Sum = 3a (d cancels instantly)
4 Terms
Optimal Symmetric Form
a - 3d, a - d, a + d, a + 3d
Common Difference
2d
Sum Simplification Benefit
Sum = 4a (d cancels instantly)
5 Terms
Optimal Symmetric Form
a - 2d, a - d, a, a + d, a + 2d
Common Difference
d
Sum Simplification Benefit
Sum = 5a (d cancels instantly)

3Sum of First n Terms of an AP

Sum of n Terms Formula

S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right] = \frac{n}{2} \left[ a + l \right]

Gauss Method for Sum of First n Natural Numbers

1
$1. \text{Write}\ S = 1 + 2 + 3 + ... + n.$
2
$2. \text{Write}\ S in \text{reverse}\ : S = n + (n-1) + ... + 1.$
3
3. Add the two equations term-by-term: 2S = (n+1) + (n+1) + ... (n times).
4
4. 2S = n(n + 1) ⇒ S = n(n + 1) / 2.

Real-life example · Finding nth Term from Given Sum Formula

If S_n = 3n² + 5n, find a_n:
• a_n = S_n - S_{n-1} = [3n² + 5n] - [3(n - 1)² + 5(n - 1)]
• a_n = 3n² + 5n - [3(n² - 2n + 1) + 5n - 5] = 3n² + 5n - [3n² - n - 2] = 6n + 2.
• First term a₁ = 6(1) + 2 = 8; Common difference d = 6.

4Exam Corner: Tips, Mistakes & Board Questions

High-Yield Board Revision Takeaways

The relation a_n = S_n - S_{n-1} is valid for all n ≥ 2; for n = 1, a₁ = S₁.
nth term from the end of an AP with last term l: a_n(end) = l - (n - 1)d.
If the sum of n terms is quadratic in n (S_n = An² + Bn), the sequence is guaranteed to be an AP with common difference d = 2A.

Exam question · CBSE Class 10 Board

Which term of the AP: 21, 18, 15... is -81?

A.28th term
B.35th term
C.34th term
D.30th term

Remember

In an AP, consecutive terms have constant common difference ==d = aₙ - aₙ₋₁==.
General term formula: ==aₙ = a + (n - 1)d==.
Sum of first n terms: ==Sₙ = (n/2)[2a + (n - 1)d] = (n/2)[a + l]==.
Sum of first n natural numbers isn(n + 1) / 2.

Check yourself

Question 1
If the common difference of an AP is 5, then what is the value of a₁₈ - a₁₃?