Arithmetic Sequences and Algebra

Question 1

What is the algebraic form of every arithmetic sequence?

Accepted Answer

x_n = an + b, where a and b are fixed numbers; a is the common difference.

Question 2

In the algebraic form x_n = an + b, what does a tell you?

Accepted Answer

a is the common difference. So for x_n = 5n - 2 the common difference is 5.

Question 3

Given x_n = 5n - 2, find the first term.

Accepted Answer

x_1 = 5 - 2 = 3. The sequence is 3, 8, 13, \ldots

Question 4

What is the algebraic form of the sum of the first n terms of an arithmetic sequence?

Accepted Answer

pn^2 + qn, where p and q are fixed numbers. There is no constant term.

Question 5

Why does the sum pn^2 + qn have no constant term?

Accepted Answer

From x_n = an + b, the sum of n terms is 	frac{1}{2}an(n+1) + bn = 	frac{1}{2}an^2 + \left(	frac{1}{2}a + bight)n: every part has a factor n.

Sample questions