Arithmetic Progressions (AP) Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Definition of Arithmetic Progression (AP):

   • An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant.
   • This constant difference is called the common difference and is usually denoted by $d$.

2. General Form of AP:

   • If $a$ is the first term and $d$ is the common difference, the $n$-th term ($T_n$) of an AP can be expressed as: $$T_n = a + (n - 1)d$$
   • Here, $n$ is the term number.

3. Sum of First $n$ Terms (S_n):

   • The sum of the first $n$ terms of an AP is given by: $$S_n = \frac{n}{2} [2a + (n - 1)d]$$
   • Alternatively, it can also be written as: $$S_n = \frac{n}{2} [a + l]$$ where $l$ is the last term of the AP.

4. Finding the $n$-th Term:

   • To find the $n$-th term of an AP: $$T_n = a + (n - 1)d$$
   • For example, if the first term $a$ is 2 and the common difference $d$ is 3, then the 5th term is: $$T_5 = 2 + (5 - 1) \times 3 = 14$$

5. Sum of the First $n$ Terms:

   • The sum of the first $n$ terms of an AP where the first term is $a$ and the last term is $l$ can be calculated as: $$S_n = \frac{n}{2} [a + l]$$
   • For example, if the first term is 2, the last term is 14, and $n$ is 5: $$S_5 = \frac{5}{2} [2 + 14] = 40$$

6. Arithmetic Mean (AM):

   • If $a_1, a_2, \ldots, a_n$ are in AP, the arithmetic mean between two terms $a_i$ and $a_j$ is the average of these terms: $$\text{AM} = \frac{a_i + a_j}{2}$$

7. Applications:

   • AP is used in various real-life applications including calculating monthly installments, saving plans, and in problems involving evenly spaced quantities.

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Formula Sheet:

1. $n$-th Term of AP:

   $$T_n = a + (n - 1)d$$
   • Where $T_n$ is the $n$-th term, $a$ is the first term, and $d$ is the common difference.

2. Sum of First $n$ Terms of AP:

   $$S_n = \frac{n}{2} [2a + (n - 1)d]$$
   • or
   $$S_n = \frac{n}{2} [a + l]$$
   • Where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $d$ is the common difference, and $l$ is the last term.

3. Common Difference:

   $$d = T_{n+1} - T_n$$
   • Where $d$ is the common difference and $T_{n+1}$ and $T_n$ are consecutive terms.

4. Arithmetic Mean:

   $$\text{AM} = \frac{a_i + a_j}{2}$$
   • Where $\text{AM}$ is the arithmetic mean between $a_i$ and $a_j$.

5. Relation Between Terms:

   • If $a, a+d, a+2d, \ldots$ is an AP, then for any term $a_n$, it holds that: $$a_n = a + (n-1)d$$