What is the formula for a linear number pattern?

The formula for a linear number pattern, also known as a linear sequence, is T_n = dn + c, where 'd' is the common difference between consecutive terms, 'n' is the term number, and 'c' is a constant.

Understanding Linear Sequences

Linear sequences are characterized by a constant difference between successive terms. This constant difference is crucial in determining the formula that defines the sequence.

The Formula Explained: T_n = dn + c

This formula is fundamental to understanding and working with linear sequences. Let's break it down:

• T_n: This represents the nth term in the sequence. For example, if you want to find the 5th term, then n = 5 and you're calculating T₅.
• d: This is the common difference between consecutive terms. You find 'd' by subtracting any term from the term that follows it.
• n: This is the term number. It indicates the position of a term within the sequence (e.g., 1st term, 2nd term, 3rd term, and so on).
• c: This is a constant. It is the value that makes the formula work for all terms in the sequence. To find 'c', you can substitute the values of the first term (T₁), the common difference 'd', and n=1 into the formula and solve for 'c'.

How to Find the Formula for a Given Linear Sequence

Here’s a step-by-step process to determine the formula for a linear sequence:

1. Identify the Common Difference (d): Calculate the difference between consecutive terms. If the difference is constant, you have a linear sequence.

   • Example: In the sequence 2, 5, 8, 11,..., the common difference is 5 - 2 = 3. So, d = 3.

2. Determine the Constant (c): Substitute the value of 'd' and the first term (T₁) into the formula T_n = dn + c. Then, solve for 'c'.

   • Example: Using the sequence 2, 5, 8, 11,..., we know d = 3 and T₁ = 2. Substituting these values:
     • 2 = 3(1) + c
     • 2 = 3 + c
     • c = -1

3. Write the Formula: Substitute the values of 'd' and 'c' into the general formula T_n = dn + c.

   • Example: For the sequence 2, 5, 8, 11,..., the formula is T_n = 3n - 1.

Example Application

Let's use the sequence 2, 5, 8, 11,... and the derived formula T_n = 3n - 1 to find the 10th term:

• T₁₀ = 3(10) - 1
• T₁₀ = 30 - 1
• T₁₀ = 29

Therefore, the 10th term in the sequence is 29.

Table Summary

Variable	Description	How to Find
Tn	nth term	Use the formula Tn = dn + c, substituting the known values of 'd', 'n', and 'c'.
d	Common Difference	Subtract any term from the term that follows it.
n	Term Number	The position of the term in the sequence (1, 2, 3, ...).
c	Constant	Substitute the values of T1, 'd', and n=1 into the formula Tn = dn + c, and solve for 'c'. Or c = T1 - d