Question

Consider the arithmetic sequence presented in the table below. What is the first term, a1, and the 22nd term of the sequence?

Answer 1

Definition: Arithmetic sequence means that you add or subtract some (same) number from the previous term to obtain the preceding one.

Example: Suppose you have the following sequence.
\(-2,~~1,~~4,~~7,~~11,~~\) (and so forth)
You will see that you add \(+3\) to each Nth term to obtain the (N+1)th term.

So, we label the Nth term as \(a_n\).
(With this definition, the first term is \(a_1\),) and the second term is \(a_2|0 and so on.)
We call this number \(3\) the difference \(d\).

Observe then, that
\(a_1+d=a_2\)
\(a_1+d+d=a_3\) or, same way \(a_1+2d=a_3\)
\(a_1+d+d+d=a_4\) or, same way \(a_1+3d=a_4\)
\(a_1+d+d+d+d=a_5\) or, same way \(a_1+4d=a_5\)

So, you will also then notice that
\(a_1+n\times d=a_n\)

And this is the recursive formula to use in general (and example is just a demonstration of this -hopefully a helpful one).

Answer 2

Fixing that line:

(With this definition, the first term is \(a_1\) and the second term is \(a_2\) and so on.)
Next, we will call \(3\) the difference \(d\).

Answer 3

So given the first term of a sequence, you can find any Nth term, using the formula.
\(a_1+nd=a_n\).