Click to Chat
1800-2000-838
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
Revision Notes On Binomial Theorem If x, y ∈ R and n ∈ N, then the binomial theorem states that (x + y)^{n} = ^{n}C_{0}x^{n} + ^{n}C_{1 }x^{n-1}y + ^{n}C_{2} x^{n-2 }y^{2} +…… … .. + ^{n}C_{r}x^{n-r }y^{r} + ….. + ^{n}C_{n}y^{n} which can be written as Σ^{n}C_{r}x^{n-r}y^{r}. This is also called as the binomial theorem formula which is used for solving many problems. Some chief properties of binomial expansion of the term (x+y)^{n}: The number of terms in the expansion is (n+1) i.e. it is one more than the index. The sum of indices of x and y is always n. The binomial coefficients of the terms which are equidistant from the starting and the end are always equal. Such an expansion always follows a simple rule which is: The subscript of C i.e. the lower suffix of C is always equal to the index of y. Index of x = n – (lower suffix of C). The (r +1)^{th}term in the expansion of expression (x+y)^{n} is called the general term and is given by T_{r+1 }= ^{n}C_{r}x^{n-r}y^{r} The term independent of x is obviously without x and is that value of r for which the exponent of x is zero. The middle term of the binomial coefficient depends on the value of n. There can be two different cases according to whether n is even or n is odd. If n is even, then the total number of terms are odd and in that case there is a single middle term which is (n/2 +1)^{th} and is given by ^{n}C_{n/2 }a^{n/2} x^{n/2}. On the other hand, if n is odd, the total number of terms is even and then there are two middle terms [(n+1)/2]^{th}and [(n+3)/2]^{th} which are equal to ^{n}C_{(n-1)/2 }a^{(n+1)/2} x^{(n-1)/2 }and ^{n}C_{(n+1)/2 }a^{(n-1)/2} x^{(n+1)/2} The binomial coefficient of the middle term is the greatest binomial coefficient of the expansion. Some results which are applied in binomial theorem problems are ^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r} ^{ n}C_{r} = n/r (^{n-1}C_{r-1}) ^{ n}C_{r}/^{n}C_{r-1} = (n – r + 1)/r [(n+1)/(r+1)] ^{n}C_{r} =^{ n+1}C_{r+1} Some of the standard binomial theorem formulas which should be memorized are listed below: C_{0} + C_{1} + C_{2} + ….. + C_{n}= 2^{n} C_{0} + C_{2} + C_{4} + ….. = C_{1} + C_{3} + C_{5} + ……….= 2^{n-1} C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + ….. + C_{n}^{2} = ^{2n}C_{n} = (2n!)/ n!n! C_{0}C_{r} + C_{1}C_{r+1} + C_{2}C_{r+2 }+ ….. + C_{n-r}C_{n} = (2n!)/ (n+r)!(n-r)! We can also replace ^{m}C_{0} by ^{m+1}C_{0} because numerical value of both is same i.e. 1. Similarly we can replace ^{m}C_{m} by ^{m+1}C_{m+1}. Note that (2n!) = 2^{n}. n! [1.3.5. … (2n – 1)] In order to compute numerically greatest term in a binomial expansion of (1+x)^{n}, find T_{r+1 }/ T_{r }= (n – r + 1)x / r. Then put the absolute value of x and find the value of r which is consistent with the inequality T_{r+1 }/ T_{r}> 1. If the index n is other than a positive integer such as a negative integer or fraction, then the number of terms in the expansion of (1+x)^{n }is infinite. The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. |x| > 1 then it is convenient to expand in powers of 1/x which is then small. The binomial expansion for the nth degree polynomial is given by: Following expansions should be remembered for |x| < 1: (1 + x)^{-1} = 1 – x + x^{2} – x^{3} + x^{4} - ….. ∞ (1 – x)^{-1} = 1 + x + x^{2 }+ x^{3} + x^{4 }+ ….. ∞ (1 + x)^{-2} = 1 – 2x + 3x^{2} – 4x^{3} + 5x^{4} - ….. ∞ (1 - x)^{-2} = 1 + 2x + 3x^{2 }+ 4x^{3} + 5x^{4}+ ….. ∞ An important result: For sums involving product of two binomial coefficients the folliwng result is used: ^{n}C_{0} ^{m}C_{k} +^{ }^{n}C_{1}^{m}C_{k-1} + ^{n}C_{2}^{m}C_{k-2 }+ …. + ^{n}C_{k}^{m}C_{0} = ^{m + n}C_{k} In order to find the integral part of the expression N = (a + √b)^{n} (n ∈ N) Step 1: Consider N’ = (a -√b)^{n} or (√b – a)^{n} according as a > √b or √b > a. Step 2: Use N + N’ or N – N’ such that the result is an integer. Step 3: Use the fact that N = I + t, where I stands for [N] and t for {N}. When n is a negative integer or a fraction then the expansion of a binomial is possible only when: (a) its first termis 1. (b) the second term is numerically less than 1. Hence, when n does not belong to n and |x| < 1, then it states that (1 + x)^{n} = 1 + nx + n(n-1)x^{2}/2! + n(n-1)(n-2)x^{3}/3! + …. n(n-1).. (n – r + 1)x^{r}/r! + … ∞ General term: T_{r+1} = n(n-1)(n-2) .. .(n – r + 1). x^{r}/r! In this expansion there are infinite terms. In this case, the differnet terms cannot be expressed as ^{n}C_{0}, ^{n}C_{1}, … because n is not a positive integer. In order to find the sum of an infinite series, we can compare the given infinite series with the expansion of (1+x)^{n} = 1 + nx + n(n-1)/2! x^{2} + … and by finding the value of x and n and putting in (1+x)^{n} the sum of series is determined.
If x, y ∈ R and n ∈ N, then the binomial theorem states that
(x + y)^{n} = ^{n}C_{0}x^{n} + ^{n}C_{1 }x^{n-1}y + ^{n}C_{2} x^{n-2 }y^{2} +…… … .. + ^{n}C_{r}x^{n-r }y^{r} + ….. + ^{n}C_{n}y^{n}
which can be written as Σ^{n}C_{r}x^{n-r}y^{r}. This is also called as the binomial theorem formula which is used for solving many problems.
The number of terms in the expansion is (n+1) i.e. it is one more than the index.
The sum of indices of x and y is always n.
The binomial coefficients of the terms which are equidistant from the starting and the end are always equal.
The subscript of C i.e. the lower suffix of C is always equal to the index of y.
Index of x = n – (lower suffix of C).
The (r +1)^{th}term in the expansion of expression (x+y)^{n} is called the general term and is given by T_{r+1 }= ^{n}C_{r}x^{n-r}y^{r}
The term independent of x is obviously without x and is that value of r for which the exponent of x is zero.
The middle term of the binomial coefficient depends on the value of n. There can be two different cases according to whether n is even or n is odd.
If n is even, then the total number of terms are odd and in that case there is a single middle term which is (n/2 +1)^{th} and is given by ^{n}C_{n/2 }a^{n/2} x^{n/2}.
On the other hand, if n is odd, the total number of terms is even and then there are two middle terms [(n+1)/2]^{th}and [(n+3)/2]^{th} which are equal to ^{n}C_{(n-1)/2 }a^{(n+1)/2} x^{(n-1)/2 }and ^{n}C_{(n+1)/2 }a^{(n-1)/2} x^{(n+1)/2}
The binomial coefficient of the middle term is the greatest binomial coefficient of the expansion.
Some results which are applied in binomial theorem problems are
^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}
^{ n}C_{r} = n/r (^{n-1}C_{r-1})
^{ n}C_{r}/^{n}C_{r-1} = (n – r + 1)/r
[(n+1)/(r+1)] ^{n}C_{r} =^{ n+1}C_{r+1}
Some of the standard binomial theorem formulas which should be memorized are listed below:
C_{0} + C_{1} + C_{2} + ….. + C_{n}= 2^{n}
C_{0} + C_{2} + C_{4} + ….. = C_{1} + C_{3} + C_{5} + ……….= 2^{n-1}
C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + ….. + C_{n}^{2} = ^{2n}C_{n} = (2n!)/ n!n!
C_{0}C_{r} + C_{1}C_{r+1} + C_{2}C_{r+2 }+ ….. + C_{n-r}C_{n} = (2n!)/ (n+r)!(n-r)!
We can also replace ^{m}C_{0} by ^{m+1}C_{0} because numerical value of both is same i.e. 1. Similarly we can replace ^{m}C_{m} by ^{m+1}C_{m+1}.
Note that (2n!) = 2^{n}. n! [1.3.5. … (2n – 1)]
In order to compute numerically greatest term in a binomial expansion of (1+x)^{n}, find T_{r+1 }/ T_{r }= (n – r + 1)x / r. Then put the absolute value of x and find the value of r which is consistent with the inequality T_{r+1 }/ T_{r}> 1.
If the index n is other than a positive integer such as a negative integer or fraction, then the number of terms in the expansion of (1+x)^{n }is infinite.
The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. |x| > 1 then it is convenient to expand in powers of 1/x which is then small.
The binomial expansion for the nth degree polynomial is given by:
Following expansions should be remembered for |x| < 1:
(1 + x)^{-1} = 1 – x + x^{2} – x^{3} + x^{4} - ….. ∞
(1 – x)^{-1} = 1 + x + x^{2 }+ x^{3} + x^{4 }+ ….. ∞
(1 + x)^{-2} = 1 – 2x + 3x^{2} – 4x^{3} + 5x^{4} - ….. ∞
(1 - x)^{-2} = 1 + 2x + 3x^{2 }+ 4x^{3} + 5x^{4}+ ….. ∞
An important result:
For sums involving product of two binomial coefficients the folliwng result is used:
^{n}C_{0} ^{m}C_{k} +^{ }^{n}C_{1}^{m}C_{k-1} + ^{n}C_{2}^{m}C_{k-2 }+ …. + ^{n}C_{k}^{m}C_{0} = ^{m + n}C_{k}
In order to find the integral part of the expression N = (a + √b)^{n} (n ∈ N)
Step 1: Consider N’ = (a -√b)^{n} or (√b – a)^{n} according as a > √b or √b > a.
Step 2: Use N + N’ or N – N’ such that the result is an integer.
Step 3: Use the fact that N = I + t, where I stands for [N] and t for {N}.
When n is a negative integer or a fraction then the expansion of a binomial is possible only when:
(a) its first termis 1.
(b) the second term is numerically less than 1.
Hence, when n does not belong to n and |x| < 1, then it states that
(1 + x)^{n} = 1 + nx + n(n-1)x^{2}/2! + n(n-1)(n-2)x^{3}/3! + …. n(n-1).. (n – r + 1)x^{r}/r! + … ∞
General term:
T_{r+1} = n(n-1)(n-2) .. .(n – r + 1). x^{r}/r!
In this expansion there are infinite terms.
In this case, the differnet terms cannot be expressed as ^{n}C_{0}, ^{n}C_{1}, … because n is not a positive integer.
In order to find the sum of an infinite series, we can compare the given infinite series with the expansion of (1+x)^{n} = 1 + nx + n(n-1)/2! x^{2} + … and by finding the value of x and n and putting in (1+x)^{n} the sum of series is determined.
Post Question
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.