Lesson 8 And 12 Discrete Mathematics /100% Correct
Mathematics homework help
(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