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# Binomial Theorem

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### Binomial Theorem

1. 1. BINOMIAL THEOREM
2. 2. BINOMIAL THEOREM FOR POSITIVE INTEGRALEXPONENTWhen n is a positive integer, then n n x y C0 x n n C1 x n 1 y n C2 x n 2 y2 .... n Cr x n r yr ... n Cn y n
3. 3. PROPERTIES OF BINOMIAL EXPANSION: The number of terms in the expansion is n + 1. The sum of the indices of x and y in each term is n. The binomial coefficients (nC0, nC1 ..........nCn) of the terms equidistant from the beginning and the end are equal, i.e. nC = nC , nC = n C n n 0 n 1 n–1 etc. { Cr = Cn–r}
4. 4. SOME IMPORTANT TERMS IN THE EXPANSION OF (X + Y)N :  General term : (x + y)n = nC0 xn y0 + nC1 xn–1 y1 + ...........+ nCr xn–r yr +..........+ nCn x0 yn (r + 1)th term is called general term. Tr+1 = nCr xn–r yr  Middle term(s) : I. If n is even, there is only one middle term, which is n 2 th term. 2 II. If n is odd, there are two middle terms, which are n 1 2 th term n 1 1th term. 2
5. 5. NUMERICALLY GREATEST TERM Numerically greatest term in the expansion of (x + y)n, n Î N Let Tr and Tr+1 be the rth and (r + 1)th terms respectively Tr = nCr–1 xn–(r–1) yr–1 Tr+1 = nCr xn–r yrFor term Tr+1 to be numerically greatest Tr < Tr+1> Tr+2
6. 6. PROPERTIES OF BINOMIAL COEFFICIENTS nIf 1 x C0 C1x C2 x 2 .... Cn x n , then :i. C0 C1 C2 .... Cn 2n Sum of binomial coefficients is 2n.ii. C1 C2 C3 ... 2n 1iii. C0 C2 C4 ... C1 C3 C5 ... 2n 1 2 2 2 2 2niv. C0 C1 C2 ... Cn Cn 2nv. C0Cr C1Cr 1 C2Cr 2 ... Cn r Cr Cn 1, n where C0 ,C1 ,C2, …….represent C0 , n C1 , n C2 ,....
7. 7. GREATEST COEFFICIENTS In (x + y)n, the binomial coefficients are n C0 , n C1 , n C2 ,.... n Cn Here coefficient in tr+1 is nCr . If n is even, nCr is greatest when r n/2 . If n is odd, nCr is greatest when r (n+1)/2 .
8. 8. GREATEST TERM We follow the following steps in order to the find the greatest term (numerically) in the expansion of (1 x)n. x n 1 Find p x 1 If p is an integer, then tp and tp+1 are equal and each is the greatest term. If p is not an integer, then t[p]+1 is the greatest term, where [.] denotes the greatest integral part. To find the greatest term in the expansion of (x + y)n. n n n y x y x 1 x n Now find the greatest term in y. 1 x
9. 9. EXAMPLES
10. 10. MULTINOMIAL EXPANSION If n N , then the general term in the multinomial n expansion of x1 x 2 ... x k is: n! a x11 x 32 x 33 ...x a k a a k a1 !a 2 !a 3 !...a k ! where a1 + a2 + a3 +…….+ ak = n and 0 a i n,i 1,2,3..., k and the number of terms in the n k 1 expansion is Ck 1 .
11. 11. BINOMIAL THEOREM FOR RATIONALEXPONENTS STATEMENT If n is rational number, then : n n n 1 1 x 1 nx x 2 ... 2! n n 1 n 2 .... n r 1 x r ...to r! where |x| < 1 . n n 1 n 2 ... n r 1 General term tr 1 xr r!
12. 12. KEY POINTSi. The exponent of x in the first term is the same as that of Binomial.ii. The exponent of y in the first term is zero.iii. In each successive term, the exponent of x goes on decreasing by 1 and of y goes on increasing by 1.iv. The sum of exponents of x and y in each term is n, the exponent of the binomial.v. Total number of terms = n+1 i.e. one more than the power of the binomial.vi. General item, n tr 1 Cr x n r y r
13. 13. i. Sum of the coefficients of odd terms = Sum of coefficients of even terms = 2n-1.ii. Coefficients of terms equidistant from the beginning and end are equal. When n is even, n Cn / 2 is greatest coefficient. When n is odd, n n Cn 1 or Cn 1 is the greatest coefficient. 2 2
14. 14. EXAMPLES
15. 15. Particular Cases : 1I. 1 x 1 x x2 x 3 ....to 1II. 1 x 1 x x2 x 3 ....to 2III. 1 x 1 2x 3x 2 4x 3 ....to 2IV. 1 x 1 2x 3x 2 4x 3 ......to