Revision Notes: Binomial Theorem & its Simple Applications

# Revision Notes: Binomial Theorem & its Simple Applications | Mathematics (Maths) for JEE Main & Advanced PDF Download

## Revision Notes On Binomial Theorem

• If x, y ∈ R and n ∈ N, then the binomial theorem states that  (x+y)n = nC0xn + nCxn-1y+ nC2 xn-2 y2 +…… … .. + nCrxn-r yr + ….. + nCnywhich can be written as ΣnCrxn-ryr. 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. The simple reason behind this is C(n, r) = C(n, n-r) which gives C(n, n) C(n, 1) = C(n, n-1) C(n, 2) = C(n, n-2).
• 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)thterm in the expansion of expression (x+y)n is called the general term and is given by Tr+1 = nCrxn-ryr

• 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 thevalue 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 nCn/2 an/2 xn/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]thand [(n+3)/2]th which are equal to nC(n-1)/2 a(n+1)/2 x(n-1)/2 and nC(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 of the standard binomial theorem formulas which should be memorized are listed below:

• C0 + C1 + C2 + ….. + Cn= 2n
• C0 + C2 + C4 + ….. = C1 + C3 + C5 + ……….= 2n-1
• C02 + C12 + C22 + ….. + Cn= 2nCn = (2n!)/ n!n!
• C0Cr + C1Cr+1 + C2Cr+2+ ….. + Cn-rCn=(2n!)/ (n+r)!(n-r)!
• Another result that is applied in binomial theorem problems is nCr + nCr-1 = n+1Cr
• We can also replace mC0 by m+1C0 because numerical value of both is same i.e. 1. Similarly we can replace mCm by m+1Cm+1.
• Note that (2n!) = 2nn! [1.3.5. … (2n-1)]
• In order to compute numerically greatest term in a binomial expansion of (1+x)nfind Tr+1 / Tr= (n – r + 1)x/r. Then put the absolute value of x and find the value of r which is consistent with the inequality Tr+1 / Tr> 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)nis 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 + x2 – x3 + x4 - ….. ∞
• (1-x)-1 = 1 + x + x2+ x3 + x4+ ….. ∞
• (1+x)-2 = 1 – 2x + 3x2 – 4x3 + 5x4 - ….. ∞
• (1-x)-2 = 1 +2x + 3x2+4x3 + 5x4+ ….. ∞
The document Revision Notes: Binomial Theorem & its Simple Applications | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## Mathematics (Maths) for JEE Main & Advanced

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