(x+a)

^{n}

=

^{n}C

_{0}x

^{n}a

^{0}+

^{n}C

_{1}x

^{n-1}a

^{1}+

^{n}C

_{2}x

^{n-2}a

^{2}+ ...+

^{n}C

_{r}x

^{n-r}a

^{r}+ ...+

^{n}C

_{n-1}x

^{1}a

^{n-1}+

^{n}C

_{n}x

^{0}a

^{n}

(x+a)

^{n}

= (r = 0 to n)Σ

^{n}C

_{r}x

^{n-r}a

^{r}

Some conclusions

1. total number of terms in the expansion = n+1

2. The sum of indices of x and a in each term is n.

3. the coefficients of terms equidistant from the beginning and the end are equal.

4. (x-a)

^{n}

= (r = 0 to n)Σ ((-1)

^{r}

_{*}

^{n}C

_{r}x

^{n-r}a

^{r}

The terms in the expansion of(x-a)

^{n}are alternatively positive and negative, the last term is positive or negative according as n is even or odd.

5. (1+x)

^{n}= (r = 0 to n)Σ

^{n}C

_{r}x

^{r}

you get it by putting x =1 and a = x in the expression for (x+a)

^{n}.

6. (x+1)

^{n}= (r = 0 to n)Σ

^{n}C

_{r}x

^{n-r}

7. (1-x)

^{n}= (r = 0 to n)Σ(-1)

^{r}

_{*}

^{n}C

_{r}x

^{r}

8. (x+a)

^{n}+(x-a)

^{n}= 2[

^{n}C

_{0}x

^{n}a

^{0}+

^{n}C

_{2}x

^{n-2}a

^{2}+

^{n}C

_{4}x

^{n-4}a

^{4}+ ...]

9. General term in a binomial expansion

(r+1) term in (x+a)

^{n}

=

^{n}C

_{r}x

^{n-r}a

^{r}

10. Another form of binomial expansion

(x+a)

^{n}

= (r = 0 to n) and r+s = n Σ (n!/r!s!) x

^{r}a

^{s}

11. Coefficient of (r+1)th term in the binomial expression of (1+x)

^{n}is

^{n}C

_{r}

12. Algorithm to find the greatest term

(i) Write term r+1 = T(r+1) and term r = T® from the given expression.

(ii) Find T(r+1)/T®

(iii) Put T(r+1)/T®>1

(iv) Solve the inequality for r to get an inequality of the form r

If m is an integer, the mth and (m+1)th terms are equal in magnitude and these two are the greatest terms.

If m is not an integer, then obtain integral part of m, say, k. In this case (k+1) term is the greatest term

13 Properties binomial coefficients

(i) the sum of binomial coefficients in the expansion of (1+x)

^{n}is 2

^{n}

(ii) the sum of odd binomial coefficients in the expansion of (1+x)

^{n}is equal to the sum of the coefficients of even terms and each is equal to 2

^{n-1}.

14. Middle terms in binomial expression

If n is even the (n/2 +1) th term is middle term.

If n is odd then ((n+1)/2) th and ((n=3)/2)th terms are two middle terms

**Multinomial theorem**

(x1+x2)

^{n}= (r1 = 0 to n) and r1+r2 = nΣ (n!/r1!r2!) x1

^{r1}x2

^{r2}

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