Arithmetic Progressions

Concept 1: Definition of Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant.

• Hindi: Ganitik Ardh Pragati (GAP)

Key Definitions

• First Term (a): The first term of an arithmetic progression.
• Common Difference (d): The constant difference between consecutive terms in an arithmetic progression.
• n-th Term (an): The nth term of an arithmetic progression.
• Sum of n Terms (Sn): The sum of the first n terms of an arithmetic progression.

Key Formulas

Formula 1: n-th Term of an Arithmetic Progression

an = a + (n - 1)d

Where:

• an is the n-th term
• a is the first term
• n is the term number
• d is the common difference

Formula 2: Sum of n Terms of an Arithmetic Progression

Sn = (n / 2) (2a + (n - 1)d)

Where:

• Sn is the sum of n terms
• n is the number of terms
• a is the first term
• d is the common difference

Formula 3: Sum of n Terms using Arithmetic Mean

Sn = n (a + an) / 2

Where:

• Sn is the sum of n terms
• n is the number of terms
• a is the first term
• an is the n-th term

Important Relationships

• an = a + (n - 1)d
• an + an+1 = 2an + d

Solved Examples

• Find the 10th term of an arithmetic progression given the first term 'a' is 5 and the common difference 'd' is 3. an = a + (n - 1)d = 5 + (10 - 1)3 = 32
• Find the sum of 10 terms of an arithmetic progression given the first term 'a' is 6 and the common difference 'd' is 2. Sn = (n / 2) (2a + (n - 1)d) = (10 / 2) (2(6) + (10 - 1)2) = 285

Practice Questions

• Find the 15th term of an arithmetic progression given the first term 'a' is 2 and the common difference 'd' is 4.
• Find the sum of 12 terms of an arithmetic progression given the first term 'a' is 8 and the common difference 'd' is 5.