Binomial-theorem.pdf 1a4p5x

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Binomial Theorem 1.

If n is a positive integer then ( x + a)n = n C 0 x n + nC1x n −1a + n C 2 x n − 2 a 2 + ... + nCr x n − r a r + ...... + nCna n .

2.

The expansion of ( x + a)n contains (n + 1) .

3.

In the expansion, the sum of the powers of x and a in each term is equal to n.

4.

In the expansion, the coefficients n C 0 , nC1, n C 2 ,...,n Cn are called binomial coefficients and these are simply denoted by C0 , C1, C 2 ,..., Cn .

5.

In the expansion, (r + 1)th term is called the general term. It is denoted by Tr +1 . Thus Tr +1 = n Cr x n − r ar .

6.

( x + a )n =

n

∑

n

Cr x n − r a r .

r =0

n

7.

( x − a )n =

∑

n

Cr x n − r ( −a)r =

r =0

n

∑ (−1)

n n

Cr x n − r ar = n C 0 x n − n C1x n −1a + n C 2 x n − 2 a 2 − ... + ( −1)n n Cn a n

r =0

n

8.

(1 + x )n =

∑

n

Cr x r = n C 0 + n C1x + n C 2 x 2 + ... + n Cn x n = C 0 + C1x + C 2 x 2 + ... + Cn x n

r =0

9.

Middle term(s) in the expansion of (x + a)n : n i) If n is even then ⎛⎜ + 1⎞⎟ th term is the middle term. ⎝2

ii) If n is odd then

⎠

n+1 n+3 th and th are the middle . 2 2

10. Numerically greatest term in the expansion of (1 + x)n : i) If

(n + 1) | x | = | x | +1

p, a positive integer then pth and (p+1)th are the numerically greatest

in the expansion of (1 + x)n. ii) If

(n + 1) | x | = | x | +1

p+F where p is a positive integer and 0 < F < 1 then (p+1)th term is the

numerically greatest term in the expansion of (1+x)n. 11. If Cr denotes n Cr then i) C0 + C1 + C 2 + ... + Cn = 2n ii) C0 + C 2 + C 4 + ... = C1 + C3 + C 5 + ... = 2n −1 n

12. i)

∑ r =0

n

n

Cr = 2n

ii)

∑ (−1)

r n

Cr = 0

r =0

13. a⋅C0 + (a + d)⋅C1 + (a + 2d)⋅C2 +……+ (a + nd)⋅Cn = (2a + nd) ⋅2n–1. 14.

C1 + 2 ⋅ C 2 x + 3 ⋅ C 3 x 2 + ... + n ⋅ C n x n −1 = n(1 + x) n −1

Binomial Theorem 15. i) C1 + 2 ⋅ C 2 + 3 ⋅ C3 + ... + n ⋅ Cn = n ⋅ 2n −1 ii) C1 − 2 ⋅ C 2 + 3 ⋅ C 3 − ...... = 0 n

16. i)

∑ r.

n

n

Cr = n.2n −1

ii)

∑ (−1) r. r

n

Cr = 0

r =0

r =0

C C1 C (1 + x) n +1 − 1 x + 2 x 2 + .... + n x n = . n +1 2 3 (n + 1) x

17.

C0 +

18.

C 0 C r + C1C r +1 + C 2 C r + 2 + ..... + C n−r C n =

19.

C 02 + C12 + C 22 + ... + Cn2 =

(2n)! (n! ) 2

2n

C n −r

=

(2n)! . (n − r )! (n + r )!

.

20. Binomial Theorem for rational index : If n is a rational number and |x| < 1 then 1+nx +

n(n + 1) 2 n(n − 1)(n − 2) 3 x + .... = (1 + x )n . x + 3! 2!

21. If |x| < 1 then i) (1 + x)–1 = 1 − x + x 2 − x 3 + ... + ( −1)r x r + ... ii) (1 – x)–1 = 1 + x + x 2 + x 3 + ... + x r + ... iii) (1+x)–2= 1 − 2x + 3 x 2 − 4 x 3 + ... + ( −1)r (r + 1)x r + ... iv) (1 – x)–2 = 1 + 2x + 3x 2 + 4x 3 + ... + (r + 1)x r + ... v) (1 + x ) − n = 1 − nx +

n(n − 1) 2 n(n − 1)(n − 2) 3 x + ... x − 3! 2!

vi) (1 − x ) − n = 1 + nx + −p

n(n + 1) 2 n(n + 1)(n + 2) 3 x + ... x + 3! 2!

⎛x⎞ ⎝p⎠

vii) (1 − x ) q = 1 + p⎜⎜ ⎟⎟ +

2

3

p(p + q)(p + 2q) ⎛ x ⎞ ⎜⎜ ⎟⎟ + ... 3! ⎝p⎠

p(p + q) ⎛ x ⎞ ⎜ ⎟ + 2! ⎜⎝ p ⎟⎠

22. If |x| < 1 and n is a positive integer, then i) (1 − x ) −n = 1 + nC1x + (n +1)C 2 x 2 + (n + 2)C 3 x 3 + ... ii) (1 + x ) −n = 1 − n C1x + (n +1)C 2 x 2 − (n + 2)C3 x 3 + ... n

b np − m 23. Coefficient of x m in the expansion of ⎛⎜ ax p + q ⎞⎟ is the coefficient of Tr +1 where r = . ⎝

p+q

x ⎠

n

b np 24. The term independent of x in the expansion of ⎛⎜ ax p + q ⎞⎟ is Tr +1 where r = . ⎝

25. In the expansion ( x + a)n ,

x ⎠

p+q

Tr +1 n − r + 1 a . = Tr r x

26. The general term in the expansion of ( x1 + x 2 + ... + x p )n is n1 + n 2 + ... + np = n.

2

n! n x n1 x n2 ...x np where n1! n 2 !...np ! 1 2

Binomial Theorem 27. i) The coefficient of x n −1 in the expansion of (x – 1)(x – 2) … (x – n) is –n(n+1)/2. ii)

The coefficient of x n −1 in the expansion of (x + 1)(x + 2) … (x + n) is n(n+1)/2.

28. i) If the coefficients of pth, qth in the expansion of (1 + x)n are equal then p+q = n+2. ii) If the coefficients of x r , x r +1 in the expansion of (a + x/b)n are equal then n = (r+1)(ab+1) –1. 29. i) If the coefficients of rth, (r+1)th, (r+2)th of (1 + x)n are in A.P. then n2 – (4r+1)n + 4r2=2. ii) If the coefficients of x r −1, x r , x r +1 are in A.P. then (n – 2r)2 = n + 2. 30. If (a + b )n = I + F were I, n are positive integers, 0 < F < 1, a2 – b = 1, then i) I is an odd positive integer ii) (I + F)( I – F) = 1. 31. i) The number of in the expansion of ( x + a)n + ( x − a)n is (n+2)/2 if n is even, (n+1)/2 if n is odd. ii) The number of in the expansion of ( x + a)n − ( x − a)n is n/2 if n is even, (n+1)/2 if n is odd. 32. Sum of the coefficients of (ax + by)n is (a + b)n. 33. If f(x) = (a0 + a1x + a2x2 +…+ amxm)n then i) Sum of the coefficients = f(1). ii) Sum of the coefficients of even powers of x is

f (1) + f ( −1) . 2

iii) Sum of the coefficients of odd powers of x is

f (1) − f ( −1) . 2

2

2

2

34.

a.C 0 + (a + d).C1 + (a + 2d).C 2 + ... + (a + nd).C n2 =

35.

C 0 Cr + C1Cr + 1 + C 2 Cr + 2 + ... + Cn − r Cn = 2n Cn − r .

36.

m

C 0 n Cr + m C1 n Cr −1 + m C 2 n Cr − 2 + ... + m Cr n C 0 =

1 (2a + nd). 2n Cn . 2

(m + n)

Cr .

37. If n is a positive integer, then (1 − x ) −n = 1 + nC1x + (n + 1)C 2 x 2 + (n + 2)C 3 x 3 + ...

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