Arithmetic progression Class X CBSE class mathematics

Here is a comprehensive list of formulas for Arithmetic Progression (AP) from basic to advanced for Class X CBSE:

1. General Terms of Arithmetic Progression (AP)

• n-th Term of an AP (Tn or an):
  The formula to find the n-th term of an AP is:
  [
  a_n = a + (n – 1) \cdot d
  ]
  Where:
  ( a_n ) = n-th term
  ( a ) = first term
  ( d ) = common difference
  ( n ) = position of the term

2. Sum of First ‘n’ Terms of an AP (Sn)

• Formula 1 (when n-th term is known):
  [
  S_n = \frac{n}{2} \cdot (a + a_n)
  ]
  Where:
  ( S_n ) = sum of the first ‘n’ terms
  ( a_n ) = n-th term
• Formula 2 (when n-th term is not known):
  [
  S_n = \frac{n}{2} \cdot [2a + (n – 1) \cdot d]
  ]
  Where:
  ( S_n ) = sum of the first ‘n’ terms
  ( a ) = first term
  ( d ) = common difference
  ( n ) = number of terms

3. Common Difference (d)

The difference between consecutive terms of an AP is constant and is called the common difference.

[
d = a_{n+1} – a_n
]
Where:
( d ) = common difference
( a_{n+1} ) = (n+1)-th term
( a_n ) = n-th term

4. Sum of n terms when last term (l) is given

If the last term of an AP is given (denoted by ( l )), the sum of the first ‘n’ terms is:

[
S_n = \frac{n}{2} \cdot (a + l)
]
Where:
( S_n ) = sum of the first ‘n’ terms
( a ) = first term
( l ) = last term
( n ) = number of terms

5. Important Notes

• In an AP, the difference between any two consecutive terms is always the same.
• If the common difference ( d > 0 ), the AP is increasing, and if ( d < 0 ), the AP is decreasing.

Summary of Key Formulas:

1. n-th Term:
   ( a_n = a + (n – 1) \cdot d )
2. Sum of First ‘n’ Terms:
   ( S_n = \frac{n}{2} \cdot [2a + (n – 1) \cdot d] ) or
   ( S_n = \frac{n}{2} \cdot (a + a_n) )
3. Common Difference:
   ( d = a_{n+1} – a_n )
4. Sum with Last Term:
   ( S_n = \frac{n}{2} \cdot (a + l) )
5. Nth term from last : { l-(n-1)d}

These formulas are essential for solving any problem related to Arithmetic Progression in your Class X CBSE exams.