List the first five terms of the sequence with initial terms u 1 = 1 and u 2 = 5, and, for n ≥ 3, u n = 5u n−1 − 6u n−2

(1) What is the 50 th term of the arithmetic sequence with initial term 4 and common

difference 3?

7

(2) Evaluate

4

X

k=−3

(2k + 5). (Hint: Since there are only eight terms in the sum, you can just

write them all out and add.)

(3) Evaluate

99

X

i=0

?

− 2

3

? i

. (Hint: Since there are 100 terms in the sum, it isn’t a good idea to

write them all out and add. Use the formula for the sum of terms of a geometric sequence.

Leave the answer with exponents rather than using a calculator to try to get a decimal

approximation of the answer.)

(4) (a) List the first four terms of the sequence defined recursively by a 0 = 2, and, for n ≥ 1,

a n = 2a 2

n−1 − 1.

(b) List the first five terms of the sequence with initial terms u 1 = 1 and u 2 = 5, and, for

n ≥ 3, u n = 5u n−1 − 6u n−2 . Guess a closed form formula for the sequence. Hint: The

terms are simple combinations of powers of 2 and powers of 3.

(5) Give a recursive definition of the geometric sequence with initial term a and common

ratio r.

Hint: a n = ar n−1 isn’t a correct answer since this formula isn’t recursive. Make sure you

write down a recursive formula: (1) give the initial term, and (2) give the rule for building

new terms from previous terms.

(6) (bonus) Express in summation notation:

1

2

+

1

4

+

1

6

  • ··· +

1

2n , the sum of the reciprocals

of the first n even positive integers. (Note that there are n terms in the sum.)

Hint: A common mistake on this question is using the symbol n both as an index for

summation and to indicate the last term to be added in. To make sure you haven’t fallen

into that trap, replace every n in your formula by a specific value, say 5. The result shouldbe a sum

1

2

+

1

4

+

1

6

+

1

8

+

1

10

Strawberries

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