Arithmetic Progression Concepts

General Concepts of Arithmetic Progression

• The general formula for the ( n^{th} ) term of an arithmetic progression (AP) is given by:
  ( a_n = a + (n-1)d )
  where ( a ) is the first term, ( d ) is the common difference, and ( n ) is the term number.

• To calculate the sum of the first ( n ) terms ( (S_n) ) in an arithmetic progression, use:
  ( S_n = \frac{n}{2} [2a + (n-1)d] )
  This formula is derived from the average of the first and last terms multiplied by the number of terms.

Specific Examples

• For an AP with a first term of 5 and a common difference of 3, the 10th term can be found by substituting into the formula:
  ( a_{10} = 5 + (10-1) \cdot 3 = 5 + 27 = 32 )

Characteristics of Arithmetic Progression

• A sequence is classified as an arithmetic progression if the difference between consecutive terms is constant (common difference, ( d )).

Finding Common Difference

• Given the first term of an arithmetic progression is 2 and the 5th term is 12, the common difference can be calculated:
  ( a_5 = a + 4d ) leads to
  ( 12 = 2 + 4d )
  Solving gives ( 4d = 10 ), therefore ( d = 2.5 ).