Ms. Aayesha Ansari-Asst. Prof.
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Arithmetic Progression
Sum of series in A.P
Geometric Progression
Sum of series in G.P
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Ms. Aayesha Ansari-Asst. Prof.
Progression:
Series or sequence of a number which is written with definite
order by studying a previous number we must be able to write
the succeeding number, such arrangements of a number are
called “ Progression”
Ms. Aayesha Ansari-Asst. Prof.
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Arithmetic Progression:
• A progression increases or decreases by a constant number
then such a progression is called “ Arithmetic Progression”.
• Also, write as A.P.
• In a A.P. difference between the two terms are common or
constant.
• The difference is known as common difference and is
denoted by “d”
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Ms. Aayesha Ansari-Asst. Prof.
The nth term of A.P. is denoted by
tn= a + (n-1)d
Where,
a : first term of the progression
d : common difference
n : number of term
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Ms. Aayesha Ansari-Asst. Prof.
Let the series in A.P.
2, 4, 6, 8, 10,….. Then 50th term of this is given by
a=2 , d= 2,
n=50
tn= a + (n-1)d
t50= 2 + (50-1)2
t50 = 2 + 49x2
= 2+98
= 100
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Ms. Aayesha Ansari-Asst. Prof.
Let the series in A.P.
3, 8, 13, 18, 23,….. Then 40th term of this is given by
a=3 , d= 5,
n=40
tn= a + (n-1)d
t40= 3 + (40-1)5
t40 = 3 + 39x5
= 3+195
= 198
Ms. Aayesha Ansari-Asst. Prof.
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Series
The algebraic sum of the term of the progression or a
sequence is called Series
Sequence or progression: 1, 4, 7, 11, 14,…..
Series: 1 + 4 + 7 + 11 + 14 + ……
Sum of the first n term of the series is denoted by Sn
Sn = t1 + t2 + t3 + t4 + …. +tn
Ms. Aayesha Ansari-Asst. Prof.
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Sum of the first n term of the series is denoted by Sn
Sn = t1 + t2 + t3 + t4 + …. + tn
Sum of the first 5 term of the series is denoted by S5
S5 = t1 + t2 + t3 + t4 + t5
3, 8, 13, 18, 23,….. Then S7 term of this is given by
a=3 , d= 5, n=40 ,
∴ 𝑡1 = 3 , 𝑡2 = 8 , 𝑡3 = 13, 𝑡4 = 18 , 𝑡5 = 23 , 𝑡6 = 28 ,
𝑡7 = 33
∴ 𝑆7 = 3 + 8 + 13 + 18 + 23 + 28 + 33
𝑆7 = 126
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Sum of the series in A.P is given by
𝒏
𝑺𝒏 =
[ 𝟐𝒂 + 𝒏 − 𝟏 𝒅 ]
𝟐
Ms. Aayesha Ansari-Asst. Prof.
Consider the A.P progression 21, 27, 33, 39, 45,…. Then find 𝑺𝟐𝟎
𝒏
𝑺𝒏 =
[ 𝟐𝒂 + 𝒏 − 𝟏 𝒅 ]
𝟐
a = 21 , d = 6 , n = 20
𝟐𝟎
𝑺𝟐𝟎 =
[ 𝟐 𝐱 𝟐𝟏 + 𝟐𝟎 − 𝟏 𝟔 ]
𝟐
𝑺𝟐𝟎 = 𝟏𝟎[ 𝟒𝟐 + 𝟏𝟗 𝟔 ]
𝑺𝟐𝟎 = 𝟏𝟎[ 𝟒𝟐 + 𝟏𝟏𝟒]
𝑺𝟐𝟎 = 𝟏𝟎[ 𝟏𝟓𝟔]
𝑺𝟐𝟎 = 1560
Ms. Aayesha Ansari-Asst. Prof.
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The 4th term of A.P. is 19 and its 12th term is 51 then find its 21st
term
tn= a + (n-1)d
t4= a + (4-1)d = a + 3d
∵ t4= 19
∴ a + 3d = 19
t12= a + (12-1)d = a + 11d
∵ t12= 51
∴ a + 11d = 51
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a + 11d = 51
Ms. Aayesha Ansari-Asst. Prof.
a + 3d = 19
Subtract these two equations, we will get
8 d = 32
d=4
a = 19-3d
tn= a + (n-1)d
t21= 7 + (21-1)4
= 19 – 3 X 4
t21= 7 + (20)4
= 19 – 12
t21= 7 + 80
a=7
t21= 87
Ms. Aayesha Ansari-Asst. Prof.
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The 4th term of A.P. is 22 and its 10th term is 52 then find its 40th
term
Find the 40th term and sum of first 40 term of the sequence: -5, -2,
1, 4, 7, 10,….
Ms. Aayesha Ansari-Asst. Prof.
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Geometric Progression:
In a sequence the ratio of any term to its proceeding term is
constant, then it is called geometric Progression
3, 6, 9, 12, 15, ……
5, 10, 15, 20, 25, …
256, 128, 64, 32, 16…..
If a is the first term and r is the common ratio of G.P. Then it can
be expressed as
a, ar, ar2, ar3, ar4, ….
Ms. Aayesha Ansari-Asst. Prof.
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Ms. Aayesha Ansari-Asst. Prof.
If a is the first term and r is the common ratio of G.P. Then it can
be expressed as
a, ar, ar2, ar3, ar4, ….
And its nth term is given by
𝑡𝑛 = 𝑎𝑟 𝑛−1
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Ms. Aayesha Ansari-Asst. Prof.
Sum of the first n term of the geometric progression is given by
𝑆𝑛 =
𝑎 (1−𝑟 𝑛 )
1−𝑟
if r < 1
𝑎 (𝑟 𝑛 −1)
𝑟−1
if r > 1
OR
𝑆𝑛 =
Ms. Aayesha Ansari-Asst. Prof.
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• Find the 13th term of the 5, 20, 80, 320, 1280, …..
𝑡𝑛 = 𝑎𝑟 𝑛−1
a = 5, r = 4, n = 23
𝑡23 = 5 x 413−1
= 5 x 412
= 83886080
• Find the 12th term of the given G.P: 2, 6, 18, 54, ….
𝑡𝑛 = 𝑎𝑟 𝑛−1
a = 2, r = 3, n = 12
𝑡23 = 2 x 312−1
= 2 x 311
= 354294
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Ms. Aayesha Ansari-Asst. Prof.
• Find the 9th term of the given G.P: 1024, -512, 256,-128 ….
Answer: 4
• Find the 10th term of the given G.P: 8, 4, 2, 1, ….
Answer: 1/ 64
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Ms. Aayesha Ansari-Asst. Prof.
Find the sum up to the required term of the given G.P:
1. 1, 2, 4, 8, 6, …. ( 10th term)
2. 2, 6, 18, 54, ….. (16th term)
3. 1, ½ , ¼ , 1/8 , 1/16 , …..
𝑎 (𝑟 𝑛 −1)
𝑆𝑛 =
if r > 1
𝑟−1
1. a= 1, r = 2, n=10
𝑎 (210 −1)
𝑆𝑛 =
2−1
𝑆𝑛 =
1 x (1024−1)
1
= 1023
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Ms. Aayesha Ansari-Asst. Prof.
The 4th term of G.P. is 4 and product of 2nd and 4th term is 1. find the 6th term
The 4th and 7th term of term of G.P. are 72 and 576. Find the sum of first n term
Ms. Aayesha Ansari-Asst. Prof.
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Ms. Aayesha Ansari-Asst. Prof.
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