**Progression**

**1.2 Arithmetic progression**

**(A) Characteristics of Arithmetic Progression**

An

**arithmetic progression**is a progression in which the**difference**between any term and the immediate term before is a constant. The constant is called the**common difference,****.***d*
d = T_{n} – T_{n-1 } or d = T_{n+1} – T_{n} |

Example 1:

Example 1:

Determine whether the following number sequences is an arithmetic progression (AP) or not.

**(a)**–5, –3, –1, 1, …

**(b)**10, 7, 4, 1, -2, …

**(c)**2, 8, 15, 23, …

**(d)**3, 6, 12, 24, …

*Smart TIPS: For an arithmetic progression, you always plus or minus a fixed number*

*Solution:*

**(B) The steps to prove whether a given number sequence is an arithmetic progression**

**Step 1**: List down any three consecutive terms. [Example

*: T*

_{1}, T_{2}, T_{3}_{ }.]

**Step 2**: Calculate the values of

*T*−

_{3}*T*and

_{2}*T*−

_{2}*T*.

_{1}**Step 3**: If

*T*−

_{3}*T*=

_{2}*T*−

_{2}*T*=

_{1}*d*, then the number sequence is an arithmetic progression.

[Try Question 8 and 9 in SPM Practice 1 (Arithmetic Progression)]

Example 2:

Example 2:

Prove whether the following number sequence is an arithmetic progression

**(a)**7, 10, 13, …

**(b)**–20, –15, –9, …

Solution:Solution:

**(a)**

7, 10, 13 ← (Step 1: List down

*T*_{1}, T_{2}, T_{3}_{ })*T*–

_{3 }*T*= 13 – 10 =

_{2}**3**← (Step 2: Find

*T*–

_{3 }*T*and

_{2}*T*–

_{2}*T*)

_{1}*T*–

_{2}*T*= 10 – 7 =

_{1}**3**← (Step 2: Find

*T*–

_{3 }*T*and

_{2}*T*–

_{2}*T*)

_{1}*T*–

_{3 }*T*=

_{2}*T*–

_{2}*T*

_{1}Therefore, this

**is**an arithmetic progression.

(b)

(b)

–20, –15, –9

*T*–

_{3 }*T*= –9 – (–15) =

_{2}**6**

*T*–

_{2}*T*= –15 – (–20) =

_{1}**5**

*T*–

_{3 }*T*≠

_{2}*T*–

_{2}*T*

_{1}Therefore, this

**is not**an arithmetic progression.