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01_Straight_Line 02_Circle 03_Permutation_and_combination 04_Complex_numbers 05_Application_of_Derivative 06_Binomial_theorem 07_Progression__Series.pdf 08_Definite_Integral.pdf 09_Indefinite_Integral.pdf 10_Probability.pdf

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    BINOMIAL THEOREM BINOMIAL EXPRESSION Any algebraic expression consisting of only two terms is known as a binomial expression. BINOMIAL THEOREM Such formula by which any power of a binomial expression can be expanded in the form of a series is known as binomial theorem. For a positive integer n the expansion is given by where n Co, nC1, nC2 , r!(n—r)! Similarly (a - x) n Cn are called the binomial coefficients. The value n Cr is defined as 1-2-3...r Cia x + nC2an-2x2 + ... + ( -1 Y n Example 1 : Solution : Expand 7 1 x 7 1 x 7 cnxn 3 1 x + C2X 35 1 21 3 x 1 3 x 1 5 x 7 1 5 7 x x 1 4 x 1 + C7— 7 x x = x + + 35 x + — x GENERAL TERM IN THE EXPANSION The general term in the expansion of (a + x) n is (r + l)th term given by tr+l = nCran-W. Similarly the general term in the expansion of (x + is given by tr+l = Crxn-rar. The terms are considered from the beginning. Note: The (r + l)th term from the end = (n- r + l)th term from the beginning. The binomial coefficients in the expansion of (a + equidistant from the beginning and the end (ii) are equal. Middle term of (a + x) n (iii) th term, when n is even Example 2 : Solution : th (b) is 2 th term and term, when n is odd 2 15 3a Find the co—efficient of in +— x 15 3a General term ((r+l) th term) in + x 3a 15cr(x ) 2 15-r x 15 30-2r crx If this term contains Then 30— 24 Therefore, the co—efficient of x GREATEST BINOMIAL COEFFICIENT 15 r r 30—3r Cr3a x 3r = 24 r = 15 2 The greatest binomial coefficient is the binomial coefficient of middle term. Greatest binomial coefficient in (1 + x) n
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    Binomial Theorem is Cn/2 when nis even n Cn+l and n Cn I when nis odd 2 GREATEST TERM 2 2 where n is a To determine the numerically greatest term (absolute term) in the expansion of (a + x)n, positive integer. n Cran rxr n Cr la Thus ITr Il >ITrI if a 1 x x a x a x a —1 must be positive since n > r. Thus Tr+l will be the greatest term if r has the greatest value consistent with inequality (1 ). Example 3 : Solution: Find the greatest term in the expansion of (2 + if x = 3/2. n —r +1 3x 2 where 2 -r 90 -9r 2 10-r 9 4 r Therefore Tr+1 2 Tr if, 90 — 9r > 4r 90 > 13r 90 13 , r being an integer, hence r = 6. 313.7 = (2)3 2 PROPERTIES OF BINOMIAL COEFFICIENT For sake of convenience the coefficients respectively. Putting x = I we get + + + Putting x = Cn are usually denoted by Co, Cl, 1, we get co + + = 2 Putting x = 1 and 1 and adding, we get Co + C2+ C4 + Putting x = 1 and 1 and subtracting, we get Cl + C3 + C5 + 2/2 cos Putting x = i and equating real part, we get Co - C2 + C4 = Putting x = i and equating imaginary part, we get Cl Q + C5 4 = sin— 4
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    3 Notes: (iii) used. If (1 = E ncrxr , then prove that Example 4 : Given (l + x) n = co + C 1 x + C + + Solution: Binomial Theorem Differentiation: When the terms in an identity are the product of a numerical (natural number) and a binomial coefficient, then differentiation is used. Integration: When the numerical (natural number) occurs as the denominator of the binomial coefficient, integration is used. Multiplication of binomial expansion: When each term is summation contains the product of two binomial coefficients or square of binomial coefficient, multiplication of binomial coefficient is 2 3 Integrating w.r.t. x between the limits 0 and x we get 2 3 xn+l Cox + Cl — + C 2 — + + 2 3 2 1 3 Cox + Cl — — + 2 3 Also (1 = Coxn + I + C 2 + + Cn (2) (3) Multiplying (2) and (3) and equating coefficient of xn+l of both sides we get 2 n 4-1 cn+l -O 2 3 n +1) ! 12

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