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Title: Arithmetic Progression
Description: Basic information, important formulae and sample problems related to arithmetic progression.
Description: Basic information, important formulae and sample problems related to arithmetic progression.
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ARITHMETIC PROGRESSIONS
Definition: 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
...
10, 30, 50, 70…
...
-6, -9, -12, -15…
...
By following a certain pattern or rule, we can write the successive terms
in each of the lists above
...
In ii), each term is 20 more than the preceding term
...
e
...
In iv), each term is obtained by adding -3 to (or subtracting 3 from) the
preceding term
...
Such lists of numbers are said to be in
Arithmetic Progression (AP)
...
It can be positive, negative or zero
...
Then the AP becomes a1, a2, a3… an
...
= an – an-1
...
represents an AP where a is
the first term and d is the common difference
...
Finite AP: An arithmetic progression with finite (or fixed number of
terms)
...
Infinite AP: An arithmetic progression with infinite terms
...
The minimum information required to form an AP is the first term a and
the common difference d
...
Then the corresponding AP is:
3, 5, 7, 9, 11…
...
Given a list of numbers that form an AP, we can find the first term a and
the common difference d
...
By looking at the AP, we know the first term a = 12
...
SAMPLE PROBLEMS:
1
...
, write the first term a and the common
difference d
...
Common difference d = 1 – (-4) = 5
...
2
...
ii) -10, -8, -5, -3…
...
i)
Calculating the difference between the consecutive terms,
a2 – a1 = 15 – 5 = 10
a3 – a2 = 25 – 15 = 10
a4 – a3 = 35 – 25 = 10
Since the differences obtained are same, the given list of numbers
form an AP with common difference d = 10
...
ii)
Calculating the difference between the consecutive terms,
a2 – a1 = -8 – (-10) = 2
a3 – a2 = -5 – (-8) = 3
a4 – a3 = -3 – (-5) = 2
Since the differences obtained are not same, the given list of numbers
does not form an AP
...
an be an AP where the first term a1 is a and the common
difference is d
...
Looking at the above pattern, we can say the nth term an = a + (n – 1)d
...
Consider an AP: 4, 11, 18, 25…
...
Here a = 4, d =7, n = 8
...
Therefore the 8th term of the AP is 53
...
Find the 10th term of the AP: 2, 7, 12, 17…
...
an = a + (n-1)d
a10 = 2 + (10 -1) x 5 = 47
...
2
...
is -81?
Sol: From the given AP, a = 21, d = -3, an = -81
We are supposed to find n
...
Sum of First n Terms of an AP:
𝒏
The sum of first n terms of an AP is given by: S = [2a + (n-1)d] where
𝟐
a is the first term and d is the common difference
...
𝒏
It can be better written as: S = [a + l], replacing an with l - the last term
𝟐
of the AP
...
We
can calculate the sum even when the first and last terms of the AP are
given and the common difference is not given
...
Let’s calculate the sum of first 21 terms of the AP using the formula:
𝑛
S = [2a + (n-1)d]
2
S=
21
2
[2 x 100 + (21-1) x 50] = 12600
Therefore the sum of the first 21 terms of the AP is 12600
...
Find the sum of the first 22 terms of the AP: 8, 3, -2…
...
𝑛
22
2
2
S = [2a + (n-1)d] =
[2 x 8 + (22-1) x (-5)] = -979
Hence the sum of first 22 terms of the AP is -979
...
If the sum of first 14 terms of an AP is 1050 and its first term is 10,
find the 20th term
...
n
S = [2a + (n-1)d]
2
1050 =
14
2
[20 + 13d] = 140 + 91d
By simplification, d = 10
Hence a20 = 10 + (20-1) x 10 = 200
...
Title: Arithmetic Progression
Description: Basic information, important formulae and sample problems related to arithmetic progression.
Description: Basic information, important formulae and sample problems related to arithmetic progression.