Chapter 5: Arithematic Progressions
Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) – 3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
Each of the numbers in the list is called a term.
In (i), each term is 1 more than the term preceding it.
In (ii), each term is 30 less than the term preceding it.
In (iii), each term is obtained by adding 1 to the term preceding it.
In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding
(or subtracting) 0 to the term preceding it.
In (v), each term is obtained by adding – 0.5 to (i.e., subtracting 0.5 from) the term preceding it.
In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms.
Such list of numbers is said to form an Arithmetic Progression ( AP ).
An arithmetic progression is a list of numbers in which each term is obtained by adding a
fixed number to the preceding term except the first term.This fixed number is called the common
difference of the AP.It can be positive, negative or zero.
Let us denote the first term of an AP by $a_1$ , second term by $a_2$ , . . ., nth term by $a_n$
and the common difference by $d$.
Then the AP becomes $a_1,a_2,a_3......a_n$.
So, ${a_2-a_1=a_3-a_2=a_4-a_3=....=a_n-a_{n-1}=d}$
Some more examples of AP are:
(a) The heights ( in cm ) of some students of a school standing in a queue in the morning assembly are 147 , 148, 149, . . ., 157.
(b) The minimum temperatures ( in degree celsius ) recorded for a week in the month of January in a city, arranged in ascending order are
– 3.1, – 3.0, – 2.9, – 2.8, – 2.7, – 2.6, – 2.5
(c) The balance money ( in ₹ ) after paying 5 % of the total loan of ₹ 1000 every month is 950, 900, 850, 800, . . ., 50.
(d) The cash prizes ( in ₹ ) given by a school to the toppers of Classes I to XII are, respectively, 200, 250, 300, 350, . . ., 750.
(e) The total savings (in ₹) after every month for 10 months when ₹ 50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.
Hence,
${a, a + d, a + 2d, a + 3d, . . .}$ represents an arithmetic progression
where $a$ is the first term and $d$ the common difference.
This is called the general form of an AP.
How to find $t_n$ term in a AP
If $a$ is the $1^{st}$ term and $d$ is the common difference, then
$t_n=a+(n-1)d$
write the first term $a$ and the common difference $d$.
Since AP is respresented by ${a, a + d, a + 2d, a + 3d, . . .}$,
So, $3/2,1/2,-1/2,-3/2,.....,$ can be represented by ${a, a + d, a + 2d, a + 3d, . . .}$
So $a= 3/2$ and $d= a_n-a_{n-1}=a_2-a_1=1/2-3/2 = 1$
HomeWork: Which of the following list of numbers form an AP? If they form an AP, write the next two terms : (i) 4, 10, 16, 22, . . . (ii) 1, – 1, – 3, – 5, . . . (iii) – 2, 2, – 2, 2, – 2, . . . (iv) 1, 1, 1, 2, 2, 2, 3, 3, 3, . . . (Hint: Find the common difference $d$ for every term. If the common difference $d$ is same, the numbers are in AP)