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Important Binomial Theorem Formulas for JEE and NEET

1. BINOMIAL THEOREM : 
The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as BINOMIAL THEOREM. If  x, y ∈ R and n ∈ N, then;
Important Binomial Theorem Formulas for JEE and NEETImportant Binomial Theorem Formulas for JEE and NEET
This theorem can be proved by Induction.
OBSERVATIONS  :
(i) The number of terms in the expansion is (n + 1) i.e. one or more than the index .
(ii) The sum of the indices of x & y in each term is n  
(iii) The binomial coefficients of the terms nC0, nC1 .... equidistant from the beginning and the end are equal.

2. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE:
(i) General term
(ii) Middle term
(iii)Term independent of x &
(iv) Numerically greatest term
(i) The general term or the (r + 1)th term in the expansion of (x + y)n is given by; Tr+1 = nCr xn−r . yr
(ii) The middle term(s) is the expansion of (x + y)n is (are) :
(a) If n is even, there is only one middle term which is given by;
T(n+2)/2 = nCn/2 . xn/2 . yn/2
(b) If n is odd, there are two middle terms which are:
T(n+1)/2 & T[(n+1)/2]+1
(iii) Term independent of x contains no x ; Hence find the value of r for which the exponent  of x is zero.
(iv) To find the Numerically greatest term is the expansion of (1 + x) n, n ∈ N find Important Binomial Theorem Formulas for JEE and NEET Put the absolute value of x & find the value of r Consistent with  the inequality Important Binomial Theorem Formulas for JEE and NEET
Note that the Numerically greatest term in the expansion of (1 − x)n, x > 0, n ∈ N is the same as the greatest term in (1 + x)n.


3. If Important Binomial Theorem Formulas for JEE and NEET where I & n are positive integers, n being odd and 0 < f < 1, then (I + f). f = Kn where A − B2 = K > 0  & Important Binomial Theorem Formulas for JEE and NEET
If n is an even integer, then (I + f) (1 − f) = Kn.


4. BINOMIAL COEFFICIENTS :
(i) C0 + C1 + C2 + ....... + Cn = 2n
(ii) C0 + C2 + C4 + ....... = C1 + C3 + C5 + ....... = 2n−1
(iii) Important Binomial Theorem Formulas for JEE and NEET
(iv)Important Binomial Theorem Formulas for JEE and NEET
REMEMBER : (i) (2n)! = 2n . n! [1. 3. 5 ...... (2n − 1)]


5. BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES
If n ∈ Q, then (1 + x)n = Important Binomial Theorem Formulas for JEE and NEET
Note : (i) When the index n is a positive integer the number of terms in the expansion of (1 + x)n is finite i.e. (n + 1) & the coefficient of successive terms are:
nC0 , nC1 , nC2 , nC3 ..... nCn
(ii) When the index is other than a positive integer such as negative integer or fraction,  the number of terms in the expansion of (1 + x)n is infinite and the symbol nCr cannot be  used to denote the Coefficient of the general term.
(iii) Following expansion should be remembered (|x| < 1).
(a) (1 + x)−1 = 1 − x + x2 − x3 + x4 − .... ∞
(b) (1 − x)−1 = 1 + x + x2 + x3 + x4 + .... ∞
(c) (1 + x)−2 = 1 − 2x + 3x2 − 4x3 + .... ∞
(d) (1 − x)−2 = 1 + 2x + 3x2 + 4x3 + ..... ∞
(iv) The expansions in ascending powers of x are only valid if x is ‘small’. If x is large i.e. | x | > 1 then we may find  it convinient to expand in powers of (1/x), which then will be small.


6. APPROXIMATIONS : (1 + x)n = 1 + nx + Important Binomial Theorem Formulas for JEE and NEET
If x < 1,  the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx, approximately. This is an approximate value of (1 + x)n.


7. EXPONENTIAL SERIES :
(i) Important Binomial Theorem Formulas for JEE and NEET where x may be any real or complex &  e =

Important Binomial Theorem Formulas for JEE and NEET
(ii) Important Binomial Theorem Formulas for JEE and NEET
Note :
(a) Important Binomial Theorem Formulas for JEE and NEET
(b) e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284.
(c) Important Binomial Theorem Formulas for JEE and NEET
(d) Important Binomial Theorem Formulas for JEE and NEET
(e) Logarithms to the base ‘e’ are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm.


8. LOGARITHMIC SERIES :
(i) Important Binomial Theorem Formulas for JEE and NEET
(ii) Important Binomial Theorem Formulas for JEE and NEET
(iii) Important Binomial Theorem Formulas for JEE and NEET
REMEMBER :
(a) Important Binomial Theorem Formulas for JEE and NEET
(b) eln x = x
(c) ln 2 = 0.693
(d) ln 10 = 2.303

The document Important Binomial Theorem Formulas for JEE and NEET is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Important Binomial Theorem Formulas for JEE and NEET

1. What is the Binomial Theorem?
Ans. The Binomial Theorem is a mathematical formula that allows us to expand expressions of the form (a + b)^n, where "a" and "b" are constants and "n" is a positive integer.
2. How is the Binomial Theorem used in mathematics?
Ans. The Binomial Theorem is used to quickly expand binomial expressions and find specific terms in the expansion without having to multiply out the entire expression manually.
3. What is the general form of the Binomial Theorem?
Ans. The general form of the Binomial Theorem is (a + b)^n = C(n,0)*a^n*b^0 + C(n,1)*a^(n-1)*b^1 + ... + C(n,k)*a^(n-k)*b^k + ... + a^0*b^n, where C(n,k) represents the binomial coefficient.
4. How do you calculate the binomial coefficient in the Binomial Theorem?
Ans. The binomial coefficient C(n,k) can be calculated using the formula C(n,k) = n! / (k! * (n-k)!), where "n!" represents the factorial of "n" and "k!" represents the factorial of "k".
5. Can the Binomial Theorem be used for negative exponents?
Ans. Yes, the Binomial Theorem can be extended to negative exponents using the formula (a + b)^(-n) = 1 / (a^n * (1 + b/a)^n), where "a" and "b" are constants and "n" is a positive integer.
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