Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

This will help you to progress in progression, Cheat Sheet of Mathematics

Mathematics

The most irritating chapter of mahs has been now brought to you on a single piece of paper

Typology: Cheat Sheet

2025/2026

Uploaded on 06/06/2026

test-user-11
test-user-11 🇮🇳

12 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
•
S
u
m
o
f
a
l
l
t
e
r
m
s
i
n
A
P
o
f
c
o
m
m
o
n
t
e
r
m
s
=
2
(
2
9
+
(
n
-
1
)
d
)
•
W
h
e
n
e
v
e
r
a
s
u
m
o
f
f
i
r
s
t
n
t
e
r
m
s
o
f
a
n
A
P
a
r
e
i
n
f
o
r
m
o
f
a
q
u
a
d
r
a
t
i
c
S
n
=
A
n
z
+
B
n
d
=
2
A
a
,
=
A
+
B
G
E
O
M
E
T
R
I
C
P
R
O
G
R
E
S
S
I
O
N
:
r
a
t
i
o
o
f
a
n
y
t
e
r
m
a
n
d
i
t
'
s
p
r
e
c
e
d
i
n
g
i
t
i
s
c
o
n
s
t
a
n
t
9
t
h
=
9
9
^
-
1
9
,
a
r
,
9
m
²
.
.
.
a
n
"
-
1
i
s
i
n
G
P
I
f
a
i
s
t
h
e
f
i
r
s
t
t
e
r
m
o
f
t
h
e
G
P
r
=
c
o
m
m
o
n
r
a
t
i
o
T
h
e
G
P
i
s
f
i
n
i
t
e
a
n
d
c
o
n
s
i
s
t
s
o
f
m
t
e
r
m
s
:
.
n
t
h
t
e
r
m
f
r
o
m
t
h
e
e
n
d
=
a
r
m
-
n
n
t
h
t
e
r
m
f
r
o
m
t
h
e
e
n
d
o
f
a
G
P
w
i
t
h
l
a
s
t
t
e
r
m
l
a
n
d
c
o
m
m
o
n
M
a
t
o
9
=
I
n
-
1
i
S
u
m
o
f
n
t
e
r
m
s
i
n
G
P
:
a
(
9
^
-
1
)
;
9
#
1
9
-
1
w
h
e
n
9
=
1
:
n
a
i
f
l
i
s
t
h
e
l
a
s
t
t
e
r
m
o
f
t
h
e
G
P
:
l
r
-
a
M
-
1
S
u
m
o
f
i
n
f
i
n
i
t
e
G
P
:
♀
,
;
1
9
1
<
1
:
0
;
1
9
1
>
1
P
r
o
p
e
r
t
i
e
s
:
i
f
9
,
,
9
2
,
a
s
.
.
.
a
r
e
i
n
G
P
i
]
K
a
i
,
K
a
z
,
K
a
z
.
.
.
a
r
e
i
n
G
P
i
i
)
l
♀
.
.
#
i
#
.
.
-
a
r
e
i
n
G
P
"
1
)
a
t
,
'
a
t
i
f
.
.
.
a
r
e
i
n
G
p
2
i
f
9
,
.
a
z
,
9
3
.
.
.
a
r
e
i
n
G
P
]
9
,
6
1
,
9
2
6
2
,
9
3
6
3
.
_
.
a
r
e
i
n
G
P
a
n
d
6
1
,
6
2
,
6
3
.
.
.
a
r
e
i
n
G
P
"
I
9
,
1
1
½
'
%
,
a
r
e
i
n
G
P
③
i
f
a
,
.
a
z
-
9
3
.
9
4
.
.
.
a
n
a
r
e
i
n
G
P
i
)
a
,
a
n
=
a
_
a
n
_
,
=
a
s
a
n
-
2
.
-
-
i
i
)
a
n
=
☑
a
n
t
h
a
m
-
n
i
i
i
]
9
,
1
9
2
,
a
s
a
r
e
i
n
G
P
o
f
n
o
n
-
z
e
r
o
a
n
d
t
h
e
n
u
m
b
e
r
:
l
o
g
a
,
,
l
o
g
a
z
,
l
o
g
a
,
.
.
.
a
r
e
i
n
A
P
i
f
f
i
r
s
t
t
e
r
m
i
s
a
a
n
d
l
a
s
t
t
e
r
m
=
L
÷
.
p
r
o
d
u
c
t
o
f
a
l
l
t
e
r
m
s
=
(
a
l
)
"
1
2
=
(
2
m
n
-
1
)
"
1
2
i
f
t
h
e
r
e
a
r
e
n
t
e
r
m
s
i
n
a
G
P
s
u
m
o
f
p
r
o
d
u
c
t
o
f
t
w
o
t
e
r
m
s
t
a
k
e
n
a
t
a
t
i
m
e
=
#
S
n
S
n
-
1
i
f
a
"
,
a
"
,
a
"
.
.
.
9
"
"
a
r
e
i
n
G
P
:
7
4
,
2
2
,
7
6
3
.
.
.
I
n
a
r
e
i
n
A
P
G
e
o
m
e
t
r
i
c
m
e
a
n
:
a
,
G
,
b
G
-
V
a
b
6
²
=
a
b
n
G
e
o
m
e
t
r
i
c
m
e
a
n
.
l
e
t
a
,
G
i
,
G
z
,
G
z
,
G
a
.
.
-
G
n
,
6
i
n
G
P
w
h
e
r
e
G
i
,
G
z
,
G
s
.
.
.
G
n
a
r
e
n
g
e
o
m
e
t
r
i
c
m
e
a
n
b
e
t
"
a
,
b
÷
6
=
n
+
2
h
t
e
r
m
÷
.
a
n
"
t
'
=
6
#
l
÷
.
M
n
t
'
=
&
:
G
-
a
(
f
)
#
G
,
G
2
G
,
G
u
.
.
.
G
n
=
(
/
9
6
)
"
a
:
(
&
)
,
G
2
=
a
(
f
)
T
H
,
.
2
G
s
=
a
(
1
)
E
H
☐
e
"
=
s
o
+
a
'
+
&
+
x
³
+
2
f
t
.
.
.
a
o
!
1
!
3
!
2
)
é
"
=
x
,
0
!
1
!
x
'
2
2
2
3
3
!
+
2
4
+
4
1
.
-
-
-
α
+
3
)
s
i
n
x
=
2
C
'
-
2
3
+
2
5
1
!
3
!
5
!
2
7
!
+
-
_
.
+
a
3
)
c
o
s
x
=
x
o
!
-
2
c
+
2
9
!
2
!
2
6
!
+
-
.
+
a
2
4
6
E
S
E
D
+
+
V
e
E
S
E
D
+
a
l
t
+
v
e
-
v
e
S
E
T
+
a
l
t
+
v
e
-
v
e
C
S
D
+
a
l
t
+
v
e
-
v
e
I
N
F
I
N
I
T
E
S
E
R
I
E
S
:
(
w
i
t
h
f
a
c
t
o
r
i
a
l
"
+
x
-
+
2
1
+
I
0
!
2
!
6
!
-
.
.
α
5
)
e
"
t
e
-
x
2
=
-
x
+
2
0
3
+
3
f
t
2
0
1
!
3
!
7
!
-
.
.
α
6
]
e
"
-
e
-
x
2
=
-
-
☐
a
"
=
e
"
/
o
g
e
a
a
"
=
I
t
x
l
o
g
e
a
+
2
7
(
l
o
g
,
a
)
²
+
,
x
!
³
(
l
o
g
_
a
)
3
.
.
.
a
s
o
;
a
#
1
]
I
N
F
I
N
I
T
E
S
E
R
I
E
S
:
(
w
i
t
h
o
u
t
f
a
c
t
o
r
i
a
l
☐
l
o
g
(
I
t
k
)
=
-
x
2
+
2
3
-
½
2
3
L
E
D
-
I
c
x
≤
I
+
l
2
)
l
o
g
(
l
-
x
)
x
-
7
c
²
x
³
.
.
.
T
-
/
E
x
c
l
2
=
3
÷
3
)
½
l
o
g
=
x
+
#
+
3
5
+
I
I
t
x
1
-
2
C
=
→
-
⅔
-
x
5
5
-
•
4
)
½
l
o
g
1
"
"
I
+
2
C
5
)
½
l
o
g
C
1
-
x
2
)
=
⟂
2
l
o
g
(
i
-
x
)
+
l
o
g
(
1
-
x
)
=
-
2
2
2
-
2
<
4
4
>
(
6
6
-
.
.
L
E
D
+
-
v
e
6
)
t
a
n
"
x
=
¼
'
-
⇒
+
I
s
_
♀
.
.
.
↓
x
⇒
7
)
I
=
1
-
I
t
4
.
.
.
8
)
l
o
g
s
=
1
-
I
t
}
-
±
.
.
.
α
9
)
t
a
n
k
=
x
-
-
3
3
+
⇐
x
5
_
I
s
a
'
I
T
E
T
+
a
l
t
-
v
e
+
v
e
-
.
T
E
T
B
I
N
O
M
I
A
L
E
X
P
A
N
S
I
O
N
:
☐
(
+
x
)
"
=
+
n
x
+
n
o
n
-
1
)
x
²
+
n
o
n
-
D
c
n
-
2
)
2
3
.
.
.
α
!
2
!
3
!
1
-
x
=
2
°
+
x
'
+
x
2
+
2
3
1
-
x
=
>
2
x
°
+
2
2
C
'
+
3
×
2
+
4
×
3
+
-
.
-
3
)
2
)
A
R
I
T
H
M
E
T
I
C
O
G
E
O
M
E
T
R
I
C
O
P
R
O
G
R
E
S
S
I
O
N
M
u
l
t
i
p
l
i
c
a
t
i
o
n
o
f
A
P
a
n
d
G
P
F
O
R
M
U
L
A
:
C
-
D
2
=
1
S
n
=
½
[
a
+
t
n
]
=
2
[
2
9
+
C
n
-
1
)
d
)
(
2
)
8
=
1
1
S
n
=
a
-
[
a
t
(
n
-
1
)
d
)
2
"
t
d
o
(
1
-
8
^
-
1
)
1
-
8
1
-
8
(
1
-
2
)
2
c
-
3
)
8
+
1
1
2
1
<
1
,
n
-
α
S
o
=
d
o
(
1
-
r
n
-
1
)
x
"
→
o
a
+
1
-
8
(
1
-
2
)
2
M
E
T
H
O
D
O
F
D
I
F
F
E
R
E
N
C
E
:
A
P
D
e
g
]
3
.
5
.
9
.
1
5
.
.
.
2
4
6
..
.
2
2
0
T
n
=
3
+
2
(
n
-
1
)
+
2
(
n
-
1
)
(
n
-
2
)
.
S
n
=
E
.
,
T
r
2
1
.
N
O
T
E
:
I
s
t
d
i
f
f
o
f
A
P
:
K
I
2
n
d
"
'
1
:
k
=
2
T
n
=
p
o
l
y
o
f
d
e
g
r
e
e
K
t
l
M
E
T
H
O
D
O
F
D
I
F
F
E
R
E
N
C
E
:
G
P
e
g
]
2
5
1
1
2
3
4
7
3
6
1
2
2
4
X
2
✗
2
✗
2
:
a
=
3
T
n
=
2
+
3
(
2
n
-
1
-
1
)
9
=
2
2
-
1
=
2
+
3
-
✗
2
n
-
3
=
3
×
2
7
-
2
2
N
O
T
E
:
I
s
t
d
i
f
f
o
f
G
P
:
K
I
2
n
d
"
'
1
:
k
=
2
T
n
=
p
o
l
y
o
f
d
e
g
r
e
e
K
-
I
+
a
r
t
'
T
n
=
2
+
(
S
n
-
,
o
f
G
P
)
T
E
L
E
S
C
O
P
I
N
G
S
E
R
I
E
S
T
y
p
e
I
:
(
U
n
s
e
r
i
e
s
)
M
u
l
T
n
b
y
E
x
)
1
.
2
.
3
+
2
.
3
.
4
+
3
.
4
.
5
t
.
.
.
+
n
t
e
r
m
s
f
i
n
d
S
n
T
h
'
=
n
o
n
t
i
)
C
n
+
2
)
n
e
x
t
f
a
c
t
o
r
-
p
r
e
v
i
o
u
s
f
a
c
t
o
r
c
o
n
s
t
a
n
t
d
i
f
f
o
f
(
N
u
m
)
=
n
(
n
t
i
)
(
n
t
2
)
[
C
n
t
3
)
-
(
n
-
1
)
]
=
⟂
4
n
C
h
t
i
)
C
n
t
2
)
(
n
+
3
)
-
C
n
-
1
)
n
(
n
t
l
)
(
n
t
z
)
S
n
=
T
+
T
+
T
s
+
T
a
+
T
l
2
5
.
.
.
(
1
-
2
-
3
.
4
-
0
)
+
(
2
.
3
.
9
.
5
-
1
-
2
-
3
.
4
)
+
(
3
.
4
-
5
-
6
-
2
.
3
.
4
.
5
)
+
n
(
n
t
l
)
(
n
t
2
)
C
n
t
3
)
T
n
¼
-
(
n
-
1
)
n
(
n
t
l
)
(
n
t
2
)
=
I
[
n
o
n
t
h
e
n
t
z
)
C
n
t
s
)
]
W
h
y
d
o
w
e
c
a
l
l
i
t
a
U
n
s
e
r
i
e
s
T
n
=
[
n
(
n
t
l
)
C
h
t
z
)
(
n
t
3
)
-
(
n
-
1
)
(
n
)
(
n
t
l
)
C
h
t
z
)
]
=
U
n
-
U
n
-
I
T
,
=
Y
-
N
o
T
2
=
V
,
T
z
=
V
i
2
-
U
3
-
V
2
T
n
=
V
n
-
V
n
-
1
[
S
n
=
V
n
-
V
o
]
n
e
x
t
f
a
c
t
o
r
-
p
r
e
v
i
o
u
s
f
a
c
t
o
r
c
o
n
s
t
a
n
t
d
i
f
f
o
f
(
N
u
m
)
T
y
p
e
2
:
R
e
c
i
p
r
o
c
a
l
o
f
T
y
p
e
1
)
M
u
l
T
n
b
y
l
a
s
t
f
a
c
t
o
r
-
f
i
r
s
t
f
a
c
t
o
r
c
o
n
s
t
a
n
t
d
i
f
f
o
f
(
N
u
m
)
E
x
]
+
+
+
.
.
.
+
1
.
2
.
3
2
.
3
.
4
3
.
4
.
5
n
t
e
r
m
s
f
i
n
d
S
n
T
n
=
[
C
h
t
z
)
-
n
]
2
n
(
n
t
l
)
(
n
+
2
)
=
½
[
n
(
n
t
l
)
-
C
a
t
l
e
n
t
z
)
+
=
±
[
#
-
±
]
T
z
=
½
[
⅓
-
±
]
T
s
=
⟂
2
[
s
a
t
-
a
t
]
÷
T
n
=
⟂
2
[
I
n
t
o
_
e
n
t
i
t
e
n
t
z
)
:
.
S
n
=
½
[
½
(
n
t
l
)
(
n
t
2
)
W
h
y
V
n
?
-
-
V
,
-
V
2
S
n
-
V
i
-
U
n
t
,
T
n
=
½
[
n
(
I
n
t
l
)
-
a
+
1
\
)
(
n
t
2
)
]
T
2
=
V
2
-
V
3
=
½
(
V
n
-
U
n
t
i
l
T
s
=
V
3
-
V
a
T
n
=
V
n
-
U
n
t
l
1
,
-
T
y
p
e
3
:
☐
9
1
-
2
.
3
+
5
+
6
2
.
3
.
4
3
.
4
.
5
+
.
.
.
n
t
e
r
m
s
T
n
=
n
+
3
n
o
n
t
i
)
(
n
+
2
)
=
n
+
2
n
o
n
t
l
)
(
n
t
2
)
+
l
n
o
n
t
i
>
c
n
t
2
)
=
n
(
n
t
l
)
+
n
C
h
t
i
)
C
n
t
2
)
T
n
,
T
n
z
S
n
,
S
n
z
+
+
S
n
=
1
2
1
-
l
1
7
1
<
1

Partial preview of the text

Download This will help you to progress in progression and more Cheat Sheet Mathematics in PDF only on Docsity!

  • Sum o f all te rm s in A P o f co m mo n t er ms = 2 ( 29 + (n- 1 ) d )
  • Wheneve r a sum of fi rs t n t erms of an A P ar e i n for m o f a q uad ra ti c

S n = A nz + Bn d = 2 A a, = A + B

G E OM ET RIC P ROGR E SS ION :

r at i o o f any t e r m and i t's pr ec eding i t is c on sta nt 9

th = 9 9 ^ - 1 9 , a r, 9 m ²... an " - 1 is in G P

If a is t he f irs t te rm o f t he GP r = co mmo n rat i o The GP is f ini t e an d c on si s ts of m ter ms :. n th te r m fr om t he e nd = ar m - n

nth ter m f ro m t h e e nd o f a GP w ith l a st t erm l a nd c o m mo n Ma to 9 = I n - 1

i

S um of n t erms i n GP : a ( 9 ^ - 1 ) ; 9 # 1 9 - 1 w h en 9 = 1 : n a if l is the las t t er m o f th e GP: l r- a M- 1

S um o f in f ini te GP : ♀ , ; 19 1 < 1

: 0 ; 1 91 > 1

P r ope rtie s: if 9 , , 9 2 , as... are in GP i] Ka i , K az , Ka z. .. ar e in G P ii)

l

♀ .. #i# ..- a re i n G P

" 1 ) a t , ' at if. .. a re in G p

(^2) if 9 ,. a z, 93 ... are in G P ] 9 , 6 1 , 926 2 , 936 3. _. a re in GP

an d 61 , 6 2 , 63 ... a^ re^ i^ n^ G^ P

"I 9 , 11 ½ ' % , are in G P

③ i f a,. az - 9 3. 94... a n a re in GP i) a , an = a_an _ , = a s an - 2. - - i i ) a n = ☑ a n t ha m- n

ii i] 9 , 19 2 , a s ar e i n GP o f n on - z er o a nd th e num b er : l oga ,, log a z, lo ga , ... ar e in AP

i f fi rs t te rm i s a an d las t te r m = L ÷. pr o duct o f all te rm s = ( a l )" 1 2 = ( 2 mn - 1 )" 1 2

i f th ere ar e n term s i n a G P su m o f pr o du ct o f two t erm s ta ken at a t ime = # Sn Sn- 1

if a " , a" , a"... 9 "" are in G P : 74 , 2 2 , 7 63. .. I n are i n AP

G eo m e t ric me a n : a , G, b G - Va b 6 ² = a b n G e o metr ic m ea n. le t a , G i , G z, G z, Ga .. - Gn , 6 in G P wh er e G i, Gz , G s. .. Gn a re n ge ome tr ic mean b e t" a, b ÷ 6 = n + 2 h t er m ÷. a n "t' = 6 ÷. Mn t ' = & :^ #^ l

G - a (f ) #

G , G 2 G , Gu... G n = ( / 9 6 )"

a:(& )

, G 2 = a ( f ) TH,.

Gs = a ( 1 )EH

☐ e" = s o + a ' + & + x ³ + 2 f t... a

o! 1! 3!

2 ) é " = x ,

0! 1!

x' 22 2 3

  • 24 + (^41).

        • α

3 ) sin x = 2 C'^ -^2 3 +^25 1! 3! 5!

27

!

  • (^) - _. + a

3 ) c os x = x o!

  • 2 c + 2

2! 9!

2

6!

  • (^) -. + a 2 4 6

ESED + + Ve

ES E D + al t + v e - v e

SE T + a l t + v e - ve

C S D + a l t + ve - ve

I N F I NI TE SE RI ES : (w it h f ac torial

" + x - + 2 1 + I

5 ) e " t e - x - .. α

2

=

x + 203 + 3 ft 20 1! 3! 7!

6 ] e " - e - x - .. α

2

=

☐ a" = e " /o ge a

a" = I t x log e a + 27 (l o g, a) ² + , x!³ ( log_ a) 3..

a so; a # 1 ].

I N F I NI TE SERI E S: (w it h ou t f a ctor i al

☐ l og ( I t k) = - x 2 + 23 - ½ 2 3

L ED

  • Ic x ≤ I

l

2 ) l og (l - x) x- 7 c² x³ ... T

  • / E xc l 2

=

3

÷

3 ) ½ l o g =^ x^ +^ #^ +^35 +

I

It x

1 - 2 C

=

→ - ⅔ -^

x 5

5

4 ) ½ l o g -^ •

1 "I + 2 "C

5 ) ½ l og C 1 - x 2 ) =

log ( i - x) + l og ( 1 - x)

= - 2 2 2 -^2 <^4 4

> ( 6 6

-..

L ED + - v e

6 ) ta n" x = ¼' - ⇒ + Is _ ♀ ...

↓ x ⇒

7 ) I =^1 - It 4 ...

8 ) lo gs = 1 - It} - ± ... α

9 ) t ank = x- - 33 + ⇐ x 5 _ Is a'

I TE T+ a l t - v e

+ ve

-. T^ ET

B IN OM IAL EX PAN S I ON:

☐ ( + x )" = + n x + n on - 1 ) x² + n o n-D c n- 2 ) 23 ... α

1 - x

= 2 ° + x ' + x 2 + 23

1 - x

= > 2

3 )^ x^ °^ +^2 2 C^ '^ +^3 ×^2 +^4 ×^3 +^ -^.^ -

ARIT H MET I CO GE OM E TR ICO PRO G R ES S IO N

Mult i pl i ca t io n o f AP an d G P

FORM ULA : C - D 2 = 1

S n = ½ [ a + tn ] = 2 [ 29 + Cn- 1 ) d )

( 2 ) 8 = 1 1

Sn = a - [ a t ( n- 1 ) d) 2 " t do ( 1 - 8 ^- 1 )

1 - 8 1 - 8 ( 1 - 2 ) 2

c - 3 ) 8 + 1 1 2 1 < 1 , n - α

S o = a + d o ( 1 - rn - 1 ) x^ "^ →^ o 1 - (^8) ( 1 - 2 ) 2

ME T H OD O F DIF F EREN CE : AP

De g] 3. 5. 9. 15...

T n = 3 + 2 (n - 1 ) + 2 ( n- 1 ) ( n- 2 )

S n = E ., Tr

(^21).

NOT E : I st dif f of A P: K I

2 nd " ' 1 : k = 2

T n = pol y of d e gr ee K tl

ME TH O D O F D I FFE REN CE : G P

eg] 2 5 1 1 23 47

3 6 12 24

X 2 ✗ 2 ✗ 2 : a = 3

Tn = 2 + 3 ( 2 n - 1 - 1 )

9 = 2

= 2 + 3 - ✗ 2 n - 3

= 3 × 2 7 - 2

2

NO T E : Ist dif f o f G P: K I

2 n d " ' 1 : k = 2

T n = p o ly o f d e g ree K - I + a rt '

Tn = 2 + ( S n - , o f G P)

TE LE SC OPIN G SE R I ES

Type I: (U n s eri es ) Mu l T n b y

Ex ) 1.^2.^3 +^2.^3.^4 +^3.^4.^5 t^.^ ..^ +^ n^ te^ rms

fi nd S n

Th' = no n ti) C n + 2 ) (^) ne xt fac to r - pre vi ous fac to r

con st a nt dif f o f (Nu m )

= n ( n ti) ( nt 2 ) [Cn t 3 ) - ( n- 1 )]

= ⟂ 4

n C ht i) C nt 2 ) ( n + 3 ) - Cn- 1 ) n ( n tl ) (nt z )

S n = T l + T 2 + T s + Ta + T 5 ...

+ n (n tl) (nt 2 ) C nt 3 )

Tn

  • ( n - 1 ) n (n tl ) ( n t 2 )

= I [non^ th^ e^ n^ tz^ )^ C^ nts)^ ]

Wh y do w e c a ll i t a Un s eri es

T n = [ n ( n tl) C ht z) ( n t 3 ) - ( n - 1 ) (n ) ( ntl ) Ch tz )]

= U n - U n - I

T, = Y - N o

T 2 = V,

Tz = V i

2 - U

3 - V 2

T n = V n - V (^) n- 1

[ Sn = V n - Vo ]

n ex t fa ct or - pre v ious fa ct or

co ns ta n t di ff of ( Nu m )

T yp e 2 : Re cipro cal of Typ e 1 ) M u l T n b y

l ast fa ct o r - f i r st f ac to r

con s tan t d i ff o f ( N um )

Ex ] + + +. .. +

    1. 3 2. 3. 4 3. 4. 5 n te rms

f i nd S n

Tn = [C ht z ) - n]

n ( nt l) ( n + 2 )^2

= ½ [ n (nt l) - Ca t l e nt z)

+ = ± [ # - ± ]

T z = ½ [ ⅓ - ± ]

T s = ⟂ 2 [ sa t - a t]

÷

T n = ⟂ 2 [ I nt o _e n ti t ent z)

:. S n = ½ [ ½ (n tl) (nt 2 )

Wh y V n?

    • V, - V 2 S n-V i - Un t, T n = ½ [ n (I nt l) - a + 1 ) ( nt 2 ) ] T 2 = V 2 - V 3

= ½ ( Vn - Unt il Ts = V 3 - Va

Tn = V n - U nt l

T y p e 3 :

☐ 9 1 - 2. 3

    1. 4 3.^4.^5
  • (^) ... (^) n t erms

Tn = (^) n + 3

non ti) (n + 2 )

= (^) n + 2

n ontl ) (n t 2 )

  • (^) l

no nt i> cn t 2 )

=

n (nt l)

n Ch ti ) Cn t 2 )

Tn, T nz

S n, Sn^ z

S n =

1 21 - l