Arithmetic Sequences and Series

In an arithmetic sequence

each successor term is the result of adding a fixed constant to . Here is a concrete example, matched up with the natural numbers:

Here the starting point is 3 and the constant is 5. From these two facts, called starting point and summand, respectively, all other terms in the sequence can be determined.

Exercise 23.2.1

Develop the recursive function a-fives, which consumes a natural number and recursively determines the corresponding term in the above series. Solution

Exercise 23.2.2

Develop the non-recursive function a-fives-closed. It consumes a natural number and determines the corresponding term in the above series. A non-recursive function is sometimes called a closed form. Solution

Exercise 23.2.3

Use series to determine the sum of the a-fives sequence for the bounds 3, 7, and 88. Can an infinite arithmetic series have a sum? Solution

Exercise 23.2.4

Develop the function seq-a-fives, which consumes a natural number n and creates the sequence of the first n terms according to a-fives or a-fives-closed. Hint: Use build-list. Solution

Exercise 23.2.5

Develop arithmetic-series. The function consumes two numbers: start and s. Its result is a function that represents the arithmetic series whose starting point is start and whose summand is s. For example, (arithmetic-series 3 5) yields a-fives (or a-fives-closed). Similarly, (arithmetic-series 0 2) produces a function that represents the series of even numbers. Solution