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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