Find the Term-to-Term Rule: Analyzing the Sequence 10, 6, 2, -2

Video Solution

Solution Steps

00:00 Find the sequence formula

00:05 Identify the first term according to the given data

00:10 Note the constant difference between terms

00:19 This is the constant difference

00:24 Use the formula for describing an arithmetic sequence

00:31 Substitute appropriate values and solve to find the sequence formula

00:45 Properly open parentheses, multiply by each factor

00:53 Continue solving

01:01 And this is the solution to the question

Step-by-Step Solution

To determine the term-to-term rule for the sequence $10, 6, 2, -2$ , we need to find the pattern in changes between terms:

Calculate the differences: $6 - 10 = -4$ , $2 - 6 = -4$ , $-2 - 2 = -4$ .
These consistent differences ( $-4$ ) indicate an arithmetic sequence with a common difference of $-4$ .

The general term of an arithmetic sequence is given by: $T_n = a + (n-1)d$ , where $a$ is the first term and $d$ is the common difference.

Substitute $a = 10$ and $d = -4$ :

$T_n = 10 + (n-1)(-4)$

$T_n = 10 - 4(n-1)$

$T_n = 10 - 4n + 4$

Simplifying gives: $T_n = 14 - 4n$

Thus, the term-to-term rule for the sequence is $14 - 4n$ .