Example 6 (a) At the beginning of this section we mentioned general sums

such as . The values to be summed are from the finite sequence

.

(b) The digits in the base-10 representation of an integer form a

finite sequence. The digit sequence of 8832 is (8, 8, 3, 2) if we take the

most significant digits first, but is (2, 3, 8, 8) if we start at the least sig-

nificant end. ■

EXERCISES 1.5

1. C alculate

(a) (b) (c)

(d) (e) (f)

2. Simplify

(a) (b)

3. Calculate

(a) for n = 1, 2, 3 and 4 (b) for n = 3, 4 and 5

(c) for n = 1, 2 and 5

4. Calculate

(a) (b) (c)

(d) (e)

5. (a) Calculate for n = 1, 2, 3, 4 and 73.

(b) Calculate for m = 1, 2 and 3. Give a formula for this product

for all m  P.

6. (a) Calculate for n = 1, 2, 3, 4 and 5.

(b) Use your answers to part (a) to guess a general formula for this sum.

7. Consider the sequence given by for n  P.

(a) List the first six terms of this sequence.

(b) Calculate for n = 1, 2, 3.

(c) Show that for n  P.

8. Consider the sequence given by for n  N.

(a) List the first seven terms of this sequence.

(b) What is its set of values?

9. For n  N, let SEQ(n) = n² — n.

(a) Calculate SEQ(n) for n  6.

(b) Show that SEQ(n + 1) = SEQ(n) + 2n for all n  N.

(c) Show that SEQ(n + 1) = SEQ(n) for n  2.

10. For n = 1, 2, 3, ... let ssq(n) = .

(a) Calculate ssq(n) for n = 1, 2, 3 and 5.

(b) Observe that ssq(n + 1) = ssq(n) + (n + 1)² for n  1.

(c) It turns out that ssq(73) = 132,349. Use this to calculate ssq(74) and ssq(72).

11. For the following sequences, write the first several terms until the behavior of the

sequence is clear.

(a) for n  N.

(b) (b_n) where b_n = a_n+1 for n  N and a_n is as in part (a).

(c) VEC(n) = (a_n, b_n) for n  N.

12. Find the values of the sequences log₂ n and for n = 16, 64, 256, and 4096, and compare.

13. (a) Using a calculator or other device, complete the table in Figure 3. [Write E if the

calculation is beyond the capability of your calculator.]

(b) Discuss the apparent relative growth behaviors of n⁴,4ⁿ, n²⁰, 20ⁿ and n!.

n	n4	4n	n20	20n	n!
5			9.54 • 1013	3.2 • 106
10				1.02 •1013	3.63 •106
25	3.91 • 105
50		1.27 • 1030