## Binomial Theorem

### Important Formulas:

       

    

   ,       

      

   ,       ,   ,  

   is 

 

 

  

 

 

 






 

16.  

17.    

18. If    is a rational number and   then       In the above series the general term is  

19. If    is a rational number and   then     

20. If    is a rational number and   then 

21. If    is a rational number and   then 

22. If    is a rational number and   then    

23. If    is a rational number and   then    

24.  If    is a rational number and   then 

25. If    is a rational number and   then 

26. If    is a rational number and   then  

27. If    is a rational number and   then 

28. If    is a rational number and   then 

29. Let  and  where, for   is the integral part of  then. If   is not an integer , then   is the numerically greatest term in the binomial expansion of 

30.Let  and  where, for   is the integral part of  then. If   is  an integer , then  are numerically greatest term in the binomial expansion of 

31.The coefficient of  in the expansion of  is  where  for all 

32.The  term independent of  in the expansion of  is  where 

33.The number of non zero terms in the expansion of  is   if  is odd

34.The number of non zero terms in the expansion of  is   if  is even.

35.The number of non zero terms in the expansion of  is   if  is even.

36.The number of non zero terms in the expansion of  is   if  is odd

37. Multinomial Theorem: If  and  then  .............1

where the summation is taken over all non negative integers    such that   and the number of terms in the expansion of 1 is 

38.If   and   is polynomial , then sum of the coefficients in  is 

39.If   and   is polynomial , then sum of the coefficients of odd power of     in  is   [sum of the coefficients of even terms]

40.If   and   is polynomial , then sum of the coefficients of odd power of     in  is   [sum of the coefficients of odd terms]

41.If the coefficients of three consecutive terms  in the expansion of   are in A.P., then 

42. The middle term in the expansion of  is 

for_BINOMIAL-THEOREM