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Mathematics - Binomial Theorem

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Revision Notes On Binomial Theorem.

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  1. Revision Notes On Binomial Theorem If x, y e R and n e N, then the binomial theorem states that 2 + nCrxn-ryr + (X+Y)n = nCoxn + nCI xn-lY+ nC2 xn- + + n Cnyn which can be written as EnCrxn-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: 2. 3. The number of terms in the expansion is (n+l) 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-l) 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). 2. The (r +1 )thterm in the expansion of expression (X+Y)n is called the general term and is given by Tr I = 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. 2. 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/2a x n/2 n/2 On the other hand, if n is odd, the total number of terms is even and then there are two middle terms which are equal to nC(nI nC(n l)2a (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: 2. 3. 4. 5. 6. 7. ..........=2nl + Cn2= 2nCn = n!n! 4. cocr + CICr+l + . Another result that is applied in binomial theorem problems is nCr + nCrI We can also replace mCO by m ICO because numerical value of both is same i.e. 1. Similarly we can replace mCm by m ICm+l Note that (2n!) = 2n. n! [1.3.5. (2n-1 In order to compute numerically greatest term in a binomial expansion of (1 +X)n, find Tr I/ T F (n — r + l)x/r. Then put the absolute value of x and find the value of r which is consistent with the inequality Tr I/ T > 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 infinite.
  2. The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. IXI > 1 then it is convenient to expand in powers of l/x which is then small. The binomial expansion for the nth degree polynomial is given by: Following expansions should be remembered for IXI < 1 : 2. 3. 2 - 1 — 2x + — 4x3 + 5e -

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