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If n is any positive integer, then
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
(v) General term in the expansion of (x + c)n is given by
(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .
(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
(xiv) If n is a positive integer, then number of terms in (x + y + z)n is (n + l)(n + 2) / 2.
Middle term in the Expansion of (1 + x)n
(i) It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.
(ii) If n is odd, then in the expansion of (x + a)n, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.
(i) If n is even, then in (x + a)n, the greatest coefficient is nCn / 2
(ii) Ifn is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn+ 1 / 2 both being equal.
In the expansion of (x + a)n
(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is Tp == Tp + 1.
(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then Tm + 1. is the greatest term.
We can conclude that,
(ii)
(iii)
(iii) The general term in the above expansion is
(iv)The greatest coefficient in the expansion of (x1 + x2 + … +
where q and r are the quotient and remainder respectively, when n is divided by m.
(v) Number of non-negative integral solutions of x1 + x2 + … + xn = n is n+ r – 1Cr – 1
Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.
0 le; f le; 1, |A – B2| = k and |√A – B| < 1
Binomial Theorem for any Index If n is any rational number, then
(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! * x r
(iii) Expansion of (x + a)n for any rational index
(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 n2 – (4r+1) n+ 4r2 = 2
(iv) In the expansion of (x + a)n
Tr + 1 / Tr = n – r + 1 / r * a / x
(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)n are equal, then p + q = n + 2
(vii) If the coefficients of xr and xr + 1 in the expansion of a + x / b)n are equal, then n = (r + 1)(ab + 1) – 1
(viii) The number of term in the expansion of (x1 + x2 + … + xr)n is n + r – 1C r – 1.
(ix) If n is a positive integer and a1, a2, … , am ∈ C,then the coefficient of xr in the expansion of
(xi) Total number of terms in the expansion of (a + b + c + d)n is (n + l)(n + 2)(n + 3) / 6.
(i) If n is a positive integer, then (1 + x)n contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)n contains infinitely many terms.
(ii) When n is a positive integer, the expansion of (l + x)n is valid for all values of x. If n is general exponent, the expansion of (i + x)n is valid for the values of x satisfying the condition |x| < 1.
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