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Binomial theorem - Reference

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Mathematics

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Academic year: 2014/2015
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Class 11 Maths Chapter 8 Binomial Theorem

Binomial Theorem for Positive Integer If n is any positive integer, then

This is called binomial theorem. Here, nC 0 , nC 1 , nC 2 , ... , nno are called binomial coefficients and nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.

Properties of Binomial Theorem for Positive Integer (i) Total number of terms in the expansion of (x + a)n is (n + 1). (ii) The sum of the indices of x and a in each term is n. (iii) The above expansion is also true when x and a are complex numbers. (iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and nCr = nCn – r, r = 0,1,2,...,n.

(v) General term in the expansion of (x + c)n is given by Tr + 1 = nCrxn – r ar. (vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease.

(vii)

(viii) (ix) The coefficient of xr in the expansion of (1+ x)n is nCr. (x)

(xi) (a)

(b) (xii) (a) If n is odd, then (x + a)n + (x – a)n and (x + a)n – (x – a)n both have the same number of terms equal to (n +1 / 2). (b) If n is even, then (x + a)n + (x – a)n has (n +1 / 2) terms. and (x + a)n – (x – a)n has (n / 2) terms. (xiii) In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term

from the beginning. (xiv) If n is a positive integer, then number of terms in (x + y + z)n is (n + l)(n + 2) / 2.

Divisibility Problems From the expansion, (1+ x)n = 1+ nC 1 x + nC 1 x 2 + ... +nCnxn We can conclude that, (i) (1+ x)n – 1 = nC 1 x + nC 1 x 2 + ... +nCnxn is divisible by x i., it is multiple of x. (1+ x)n – 1 = M(x)

(ii)

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite. (ii) General term in the expansion of (1 + x)n is Tr + 1 = n(n – 1)(n – 2)... [n – (r – 1)] / r! * xr (iii) Expansion of (x + a)n for any rational index

(vii) (1 + x)- 1 = 1 – x + x 2 – x 3 + ...∞ (viii) (1 – x)- 1 = 1 + x + x 2 + x 3 + ...∞ (ix) (1 + x)- 2 = 1 – 2x + 3x 2 – 4x 3 + ...∞ (x) (1 – x)- 2 = 1 + 2x + 3x 2 – 4x 3 + ...∞ (xi) (1 + x)- 3 = 1 – 3x + 6x 2 – ...∞ (xii) (1 – x)- 3 = 1 + 3x + 6x 2 – ...∞ (xiii) (1 + x)n = 1 + nx, if x 2 , x 3 ,... are all very small as compared to x. Important Results (i) Coefficient of xm in the expansion of (axp + b / xq)n is the coefficient of Tr + l where r = np – m / p + q (ii) The term independent of x in the expansion of axp + b / xq)n is the coefficient of Tr + l where r = np / p + q (iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)n are in AP, then n 2 – (4r+1) n

  • 4r 2 = 2 (iv) In the expansion of (x + a)n Tr + 1 / Tr = n – r + 1 / r * a / x
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Binomial theorem - Reference

Course: Mathematics

666 Documents
Students shared 666 documents in this course

University: Sikkim Manipal University

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Class 11 Maths Chapter 8 Binomial Theorem
Binomial Theorem for Positive Integer
If n is any positive integer, then
This is called binomial theorem.
Here, nC0, nC1, nC2, … , nno are called binomial coefficients and
nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.
Properties of Binomial Theorem for Positive Integer
(i) Total number of terms in the expansion of (x + a)n is (n + 1).
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex numbers.
(iv) The coefficient of terms equidistant from the beginning and the end are equal. These
coefficients are known as the binomial coefficients and
nCr = nCn – r, r = 0,1,2,…,n.
(v) General term in the expansion of (x + c)n is given by
Tr + 1 = nCrxn – r ar.
(vi) The values of the binomial coefficients steadily increase to maximum and then steadily
decrease .

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