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Binomial Theorem

1.Express the Binomial Theorem and describe it‘s properties:

Binomial Theorem and the Pascal‘s Triangle.
Binomial Theorem
For each a, b є R, n є N stands:
binomicka1a

Pascal‘s Triangle
 
binomicka1b

Stands:

binomicka1c

2. Simplify:

binomicka2
Solution:
binomicka2r

3. Simplify:

binomicka-veta-3nz

Solution:

binomicka-veta-3nr2


4.Find out the fourth member of following formula after expansion:

binomicka4
Solution:
binomicka4r

5. Find out the member of the binomial expansion of ( x + x-1)8 not containing x.

Solution:
binomicka5

The result is the number M5 = 70.

6. Which member of the binomial expansion of (2x3 + x–1)10 constains x6?

Solution:
binomicka6r

7. In the expansion of (a + 2a3)n is the coefficient of the 3. expansion member greater by 44 than the 2. member‘s coefficient.
Find out a positive integer meeting these conditions. 

Solution:
binomicka7r

8. For which x the 5th expansion member equals M5 = 105?

binomicka8
Solution:
binomicka8r