The number of terms in a binomial expansion with an exponent of n is equal to n + 1. To find a particular term in the expansion of (a + b)^{n} we make use of the general term formula.
The general term of the binomial expansion is
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Problem 1:
Without simplifying, find:
The 6^{th} term of (2x + 5)^{15}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
To find 6^{th} term,
T_{6} = T_{5+1} of expansion (2x + 5)^{15}
Putting r = 5, a = 2x, b = 5, n = 15
T_{6} = ^{15}C_{5} (2x)^{15-5} (5)^{5}
T_{6} = ^{15}C_{5} (2x)^{10} (5)^{5}
Problem 2 :
The 4^{th} term of (x² + 5/x)^{9}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
To find 4^{th} term,
T_{4} = T_{3+1} of expansion (x² + 5/x)^{9}
Putting r = 3, a = x², b = 5/x, n = 9
T_{4} = ^{9}C_{3} (x²)^{9-3} (5/x)^{3}
T_{4} = ^{9}C_{3} (x²)^{6} (5/x)^{3}
Problem 3 :
The 10^{th} term of (x - 2/x)^{17}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
To find 10^{th} term,
T_{10} = T_{9+1} of expansion (x - 2/x)^{17}
Putting r = 9, a = x, b = -2/x, n = 17
T_{10} = ^{17}C_{9} (x)^{17-9} (-2/x)^{9}
T_{10} = ^{17}C_{9} (x)^{8} (-2/x)^{9}
Problem 4 :
The 9^{th} term of (2x² - 1/x)^{21}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
To find 9^{th} term,
T_{9} = T_{8+1} of expansion (2x² - 1/x)^{21}
Putting r = 8, a = 2x², b = -1/x, n = 21
T_{9} = ^{21}C_{8} (2x²)^{21-8} (-1/x)^{8}
T_{9} = ^{21}C_{8} (2x²)^{13} (-1/x)^{8}
Problem 5 :
Find the coefficient of:
x^{10} in the
expansion of (3 + 2x²)^{10}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Putting a = 3, b = 2x², n = 10
T_{r+1} = ^{10}C_{r} (3)^{10-r} (2x²)^{r}
= ^{10}C_{r} (3)^{10-r} 2^{r }x²^{r}
To find the coefficient of x^{10}, take
x^{2r} = x^{10}
2r = 10
r = 5
T_{6 }= ^{10}C_{5} (3)^{10-5} 2^{5 }x²^{(5)}
T_{6 }= ^{10}C_{5} 3^{5} 2^{5 }x^{10}
So, coefficient of x^{10} is
^{10}C_{5} 3^{5} 2^{5}
Problem 6 :
x^{3} in the expansion of (2x² - 3/x)^{6}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Putting a = 2x², b = -3/x, n = 6
T_{r+1} = ^{6}C_{r} (2x²)^{6-r} (-3/x)^{r}
= ^{6}C_{r} (2)^{6-r }(x²)^{6-r} (-3/x)^{r}
= ^{6}C_{r} (2)^{6-r }(x)^{12-2r} (-3/x)^{r}
= ^{6}C_{r} (2)^{6-r }(x)^{12-2r} (-3)^{r }(x)^{-r}
= ^{6}C_{r} (2)^{6-r }(-3)^{r }(x)^{12-3r}
To find the coefficient of x^{3}, take
x^{12-3r} = x^{3}
12 - 3r = 3
-3r = 3 - 12
-3r = -9
r = 3
T_{4 }= ^{6}C_{3} (2)^{6-3} (-3)^{3 }x^{1}²^{-9}
T_{4 }= ^{6}C_{3} 2^{3} (-3)^{3 }x^{3}
So, coefficient of x^{3} is
^{6}C_{3} 2^{3} (-3)^{3 }
Problem 7 :
x^{12} in the expansion of (2x² - 1/x)^{12}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Putting a = 2x², b = -1/x, n = 12
T_{r+1} = ^{12}C_{r} (2x²)^{12-r} (-1/x)^{r}
= ^{12}C_{r} (2)^{12-r }(x²)^{12-r} (-1/x)^{r}
= ^{12}C_{r} (2)^{12-r }(x)^{24-2r} (-1/x)^{r}
= ^{12}C_{r} (2)^{12-r }(x)^{24-2r} (-1)^{r }(x)^{-r}
= ^{12}C_{r} (2)^{12-r }(-1)^{r }(x)^{24-3r}
To find the coefficient of x^{12}, take
x^{24-3r} = x^{12}
24 - 3r = 12
-3r = 12 - 24
-3r = -12
r = 4
T_{5 }= ^{12}C_{4} (2)^{12-4} (-1)^{4 }x²^{4-12}
T_{5 }= ^{12}C_{4} 2^{8} (-1)^{4 }x^{12}
So, coefficient of x^{12} is
^{12}C_{4} 2^{8} (-1)^{4 }
Problem 8 :
Find the constant term in:
The expansion of (x +
2/x²)^{15}
Solution:
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Putting a = x, b = 2/x², n = 15
T_{r+1} = ^{15}C_{r} (x)^{15-r} (2/x²)^{r}
= ^{15}C_{r} (x)^{15-r }2^{r} (x)^{-2r}
= ^{15}C_{r} 2^{r} (x)^{15-3r}
Constant term:
15 - 3r = 0
-3r = -15
r = 5
T_{6 }= ^{15}C_{5} 2^{5} (x)^{15-15}
T_{6 }= ^{15}C_{5} 2^{5}
So, the constant term is
^{15}C_{5} 2^{5}
Problem 9 :
The expansion of (x - 3/x²)^{9}
Solution :
General term of expansion (a + b)^{n}
T_{r+1} = ^{n}C_{r} a^{n-r} b^{r}
Putting a = x, b = -3/x², n = 9
T_{r+1} = ^{9}C_{r} (x)^{9-r} (-3/x²)^{r}
= ^{9}C_{r} (x)^{9-r }(-3)^{r} (x)^{-2r}
= ^{9}C_{r} (-3)^{r} (x)^{9-3r}
Constant term:
9 - 3r = 0
-3r = -9
r = 3
T_{4 }= ^{9}C_{3} (-3)^{3} (x)^{9-9}
T_{4 }= ^{9}C_{3} (-3)^{3}
So, the constant term is
^{9}C_{3} (-3)^{3}
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