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Chapter 7: Q. 3 (page 655)

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

An arithmetic sequence

Short Answer

Expert verified

An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference. For example, 1, 4, 7, 10, ...is an arithmetic sequence.

Step by step solution

01

Step 1. Given Information    

The given statement is an arithmetic sequence

02

Step 2. Explanation    

An arithmetic sequence is a sequence where the successive terms are either the addition or subtraction of the common term known as common difference.

The general form of arithmetic sequence is a,a+d,a+2d,a+3d,...

For example, 1, 4, 7, 10, ...is an arithmetic sequence. A series formed by using an arithmetic sequence is known as the arithmetic series for example 1 + 4 + 7 + 10... is an arithmetic series.

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Most popular questions from this chapter

Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that ifpksalmon spawn in Redfish Lake in a given year, then only 0.2pkfish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be pk+1=0.2(pk+h)..

(a) Show that the sustained number of fish returning approaches p∞=h∑k+1∞0.2kas k→∞.

(b) Evaluate p∞.

(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?

In Exercises 48–51 find all values of p so that the series converges.

∑k=1∞1(c+k)p,wherec>0

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31–48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

∑k=1∞12k+7

Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral ∫1∞a(x)dxconverges, then the nth remainder, Rn, for the series∑k=1∞a(k) is bounded by0≤Rn=∑k=n+1∞a(k)≤∫n∞a(x)dx

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

∑k=1∞ k−3/2
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