8.1. SEQUENCES
453
dollars in interest. Assume that you reinvest all interest. Then at the end of the first
month your account will contain the original principal P plus the interest, or a total of
P 1 = P + P
r
12
= P
1 +
r
12
dollars.
(a) Given that your principal is now P 1 dollars, how much interest will you earn in
the second month? If P 2 is the total amount of money in your account at the
end of the second month, explain why
P 2 = P 1
1 +
r
12
= P
1 +
r
12
2
.
(b) Find a formula for P 3 , the total amount of money in the account at the end of
the third month in terms of the original investment P.
(c) There is a pattern to these calculations. Let P n the total amount of money in
the account at the end of the third month in terms of the original investment
P. Find a formula for P n .
3. Sequences have many applications in mathematics and the sciences. In a recent paper 3
the authors write
The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of ameliorating glucose-dependent insulin secretion in subjects with diabetes. However,
its very short half-life (1.5-5 min) in plasma represents a major limitation for
its use in the clinical setting.
The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium.
For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of
GLP-1 in plasma at some time t, then only
A
2 of the hormone will be present after t + 5
minutes. Suppose A 0 = 100 grams of the hormone are initially present in plasma.
(a) Let A 1 be the amount of GLP-1 present after 5 minutes. Find the value of A 1 .
(b) Let A 2 be the amount of GLP-1 present after 10 minutes. Find the value of A 2 .
(c) Let A 3 be the amount of GLP-1 present after 15 minutes. Find the value of A 3 .
(d) Let A 4 be the amount of GLP-1 present after 20 minutes. Find the value of A 4 .
(e) Let A n be the amount of GLP-1 present after 5n minutes. Find a formula for
A n .
(f) Does the sequence { A n } converge or diverge? If the sequence converges, find
its limit and explain why this value makes sense in the context of this problem.
3 Hui H, Farilla L, Merkel P, Perfetti R. The short half-life of glucagon-like peptide-1 in plasma does not
reflect its long-lasting beneficial effects, Eur J Endocrinol 2002 Jun;146(6):863-9.
453
dollars in interest. Assume that you reinvest all interest. Then at the end of the first
month your account will contain the original principal P plus the interest, or a total of
P 1 = P + P
r
12
= P
1 +
r
12
dollars.
(a) Given that your principal is now P 1 dollars, how much interest will you earn in
the second month? If P 2 is the total amount of money in your account at the
end of the second month, explain why
P 2 = P 1
1 +
r
12
= P
1 +
r
12
2
.
(b) Find a formula for P 3 , the total amount of money in the account at the end of
the third month in terms of the original investment P.
(c) There is a pattern to these calculations. Let P n the total amount of money in
the account at the end of the third month in terms of the original investment
P. Find a formula for P n .
3. Sequences have many applications in mathematics and the sciences. In a recent paper 3
the authors write
The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of ameliorating glucose-dependent insulin secretion in subjects with diabetes. However,
its very short half-life (1.5-5 min) in plasma represents a major limitation for
its use in the clinical setting.
The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium.
For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of
GLP-1 in plasma at some time t, then only
A
2 of the hormone will be present after t + 5
minutes. Suppose A 0 = 100 grams of the hormone are initially present in plasma.
(a) Let A 1 be the amount of GLP-1 present after 5 minutes. Find the value of A 1 .
(b) Let A 2 be the amount of GLP-1 present after 10 minutes. Find the value of A 2 .
(c) Let A 3 be the amount of GLP-1 present after 15 minutes. Find the value of A 3 .
(d) Let A 4 be the amount of GLP-1 present after 20 minutes. Find the value of A 4 .
(e) Let A n be the amount of GLP-1 present after 5n minutes. Find a formula for
A n .
(f) Does the sequence { A n } converge or diverge? If the sequence converges, find
its limit and explain why this value makes sense in the context of this problem.
3 Hui H, Farilla L, Merkel P, Perfetti R. The short half-life of glucagon-like peptide-1 in plasma does not
reflect its long-lasting beneficial effects, Eur J Endocrinol 2002 Jun;146(6):863-9.