# Arithmetic Sequence Formula: Advanced General Term

In Arithmetic Sequences: General Term lesson, we saw that the general term formula is written as:

But what if we don't know the value of the first term ?

Can this formula be more generalized?

This is what we will answer in this lesson.

We know that:

If we take , so that . Similarly we get:

Subtracting equations (1) and (2) side by side:

This formula is also known as the general term of the sequence where

### Note:

Let an arithmetic sequence and a natural number with

**Example1:**

Let an arithmetic sequence in which and . Find .

**Solution:**

In this example we only know the 11th term and d. What is required is to calculate the 3rd term.(Note carefully that the first term is unknown).

To solve this problem we apply the above generalized general term property with and .

**Example2:**

Let an arithmetic sequence in which and . Find .

**Solution:**

To solve this problem we apply the above generalized general term property with and

**Example3:**

Let an arithmetic sequence in which . Find .

**Solution:**

Pay attention in this example, what is required is the calculation of , not the terms and

Let's find d first:

Thus,

**Example4:**

We consider the following arithmetic sequence:

Find and

**Solution:**

Let and .

Thus,

**Example5:**

Let an arethmitic sequence in wich

Find ,

**Solution:**

In this example, we have only one parameter, which is .

What is required is to specify and not and .

Applying the formula.

We write in terms of :

And in terms of :

Now, we adding the formula (1) and the formula (2) side by side: