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TUTORIAL 3              
GRADE : 11          BINOMIAL THEOREM & P.M.I
SUB : MATH HL                                      DUE ON 25-10-10

NAME :

1.	Given that
(1 + x)^5 (1 + ax)^6 º 1 + bx + 10x^2 + ............... + a^6 x^11,
	find the values of a, b Î *.
(Total 4 marks)

2.	Prove by induction that 12^n + 2(5^n−1) is a multiple of 7 for n Î ^+.
(Total 10 marks)

3.	Determine the first three terms in the expansion of (1− 2x)^5 (1+ x)^7 in ascending powers of x.

 (Total 5 marks)

4.	Find the coefficient of the x^3 term in the expansion of 
 (Total 6 marks)

5.	Use mathematical induction to prove that 5^n + 9^n + 2 is divisible by 4, for n Î ^+.
(Total 9 marks)

6.	(a)	Use mathematical induction to prove that
	, nÎ ^+.
(6)
(b)	Hence show that the sum of the first (n + 1) terms of the series
 +  +  +  + ... is .
(3)
(Total 9 marks)

 
7.	(a)	Use mathematical induction to prove that
(1)(1!) + (2)(2!) + (3)(3!) + ...+ (n)(n!) = (n +1)!−1 where nÎ ^+.
(8)
(b)	Find the minimum number of terms of the series for the sum to exceed 10^9.
(3)
(Total 11 marks)

8.	(a)	Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... .
(3)
(b)	Use mathematical induction to prove that for nÎ^+,
		a + ar + ar^2 + ... + ar^n–1 = 
(7)
(Total 10 marks)

2010-11/Yr 11 HL / Tut 3 – Binomial Theorem & P.M.I	Page  of