1.Algebra Booster

6.8 Algebra Booster
80. Find the sum of C 1
+ 2◊C 2
+ 3 ◊C 3
+ … + n◊C n
.
81. Find the sum of C 0
+ 2◊C 1
+ 3 ◊C 2
+ … + (n + 1)◊C n
.
82. Find the sum of 2 ◊C 1
+ 2 2 ◊C 2
+ 3 2 ◊ C 3
+ … + n 2 ◊C n
.
83. Find the sum of
C1 C2
C
C
n
0 + + + + .
2 3 n + 1
84. Find the sum of
C0 C1 C2
C
+ + + + n
.
1.2 2.3 3.4 ( n+ 1)( n+
2)
85. Prove that
m
C r
◊ n C 0
+ m C r–1
◊ n C 1
+ m C r–2
◊ n C 2
+ … + m C m
◊ n C r
= m+n C r
86. Find the sum of
C 0
◊C n
+ C 1
◊C n – 1
+ C 2
◊C n – 2
+ … + C n
◊C 0
.
87. Prove that
n
C r
◊ n C n
+ n C r–1
◊ n C n–1
+ n C r–2
◊ n C n–2
+ …
+ n C 0
◊ n C n–r
= 2n C n–r
88. Find the sum of
( n C 0
) 2 + ( n C 1
) 2 + ( n C 2
) 2 + … + ( n C n
) 2 = 2n C n
.
MULTINOMIAL THEOREM
89. Find the sum of
C 0
◊C 1
+ C 1
◊C 2
+ C 2
◊C 3
+ … + C n–1
◊ C n
.
90 Find the co-efficient of a 5 b 4 c 2` in the expansion of
(a + b + c + d) 15 .
91. Find the co-efficient of a 3 b 4 c 7 in the expansion of
(ab + bc + ca) 8 .
92. Find the greatest co-efficient of (a + b + c + d + e) 53 .
93. Find the number of terms in the expansion of
(i) (a + b + c) n
(ii) (a + b + c + d) n
(iii) (a + b + c + d + e) n .
94. Find the co-efficient of x 7 in the expansion of
(1 + 3x – 2x 3 ) 10 .
BINOMIAL THEOREM FOR ANY INDEX
95. Find the co-efficient of x n in the expansion of
1
.
1-
3x
1
96. Find the co-efficient of x n in the expansion of .
1+
4x
97. Find the co-efficient of x n in the expansion of
1
2
.
1- 9x+
20x
98. Find the co-efficient of x n in the expansion of
1
2
.
1 -( a + b)
x + abx
Ê1
+ xˆ
99. Find the co-efficient of x n in the expansion of Á
Ë1
- x˜
¯ .
Ê1
+ xˆ
100. Find the co-efficient of x n in the expansion of Á
Ë1
- x˜
¯ .
2
101. Find the co-efficient of x n in the expansion of
(1 – 2x) –1/2 .
102. Find the first negative term in the expansion of
13/3
Ê 3
1
4 x ˆ
Á +
Ë
˜
, where 0 < x < 4/3.
¯
103. If y = x – x 2 + x 3 – x 4 + …, where |x| < 1, prove that
y
x = .
1 - y
104. If y = 2x + 3x 2 + 4x 3 + …, where |x| < 1, prove that
1
x = 1 - .
1 + y
105 Find the sum of the series
1 1.3 1.3.5
1 + + + +
3 3.6 3.6.9
106. If x be so small that its square and higher powers may
be neglected, find the value of
1/2 1/4
(1 + 2 x) (16 + 3 x) 1
at x = .
2
(1 - x)
2
1/2 1/4
(1 + 2 x) + (16 + 3 x)
107 If for small values of x,
is very
1/4
(1 - 2 x)
nearly equal to a + bx, find the values of a and b.
108. If x be a quantity so small that x 3 may be neglected in
comparison of a 3 , prove that
a a 3x
+ = 2 + .
2
a + x a - x 4a
109 If p be nearly equal to q, prove that
1/9
Ê5p + 4qˆ Ê pˆ
= .
Á
Ë4p + 5q ˜
¯ Á
Ë q ˜
¯
110 If x be nearly equal to 1, prove that
m
mx - nx
m-
n
n
= x
m+
n
EXPONENTIAL SERIES
111. Find the co-efficient of x n in the expansion of e 5x .
112. Find the co-efficient of x n in the expansion of e 5x + 4 .
Ê
2
113. Find the co-efficient of x n in
1+ 3x+
2x
ˆ
Á
Ë x ˜
e ¯.
x
e
2
n
114. If B0 B1x B2x Bn
x
1 - x = + + + + , prove that
1
Bn- Bn-1
= .
n!
115. Find the value of
Ê 1 1 1
Á1
+ + + +
Ë 2! 4! 6!
Ê 1 1 1
Á1 + + + +
Ë 3! 5! 7!
.
ˆ
˜
¯
.
ˆ
˜
¯
2