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**:

**n-th Term**:

( a_n = a + (n – 1) \cdot d )**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) )**Common Difference**:

( d = a_{n+1} – a_n )**Sum with Last Term**:

( S_n = \frac{n}{2} \cdot (a + l) )**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.

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