Key to Practice Midterm III

Problem 1

A.  4,7,12,19,28,39

B.  n² + 3

Problem 2

First for n = 1, 3 = 3/2 (3 - 1) 
Assume the theorem is true for n = k-1, then
        S_i=1^k-1 3ⁱ = 3/2 (3^k-1 - 1)
We need to prove that 
        S_i=1^k 3ⁱ = 3/2 (3^k - 1)
We have that
        S_i=1^k 3ⁱ = S_i=1^k-1 3ⁱ  + 3^k 

        =  3/2 (3^k-1 - 1) + 3^k 

        = 3/2 (3^k-1 - 1) + 3/2 (2)3^k-1 

        = 3/2(3^k-1 - 1 + (2)3^k-1)

        = 3/2((3)3^k-1 - 1 )

        =  3/2((3^k - 1 )

Hence, by mathematical induction, the theorem is true.

Problem 3

S_n=1¹⁰(-1)ⁿ⁺¹(5 + 2(n - 1)/n³ 

Problem 4

699,050

Problem 5

1/2 x² - 1/6 x³ + 1/24 x⁴ 

Problem 6
-33

Problem 7
322mL

Problem 8

99,768,240

Problem 9

A.  False, this is only true for |r| < 1

B.  True, 9C4 = 126