How do you find the $12$ th term of the arithmetic sequence $20, 14, 8, 2, - 4$ …?

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Solution

Step-1: Find the common difference:
Given sequence, $20, 14, 8, 2, - 4$ .
The common difference is defined as the difference between two terms of sequence i.e. $d = a_{2} - a_{1}$
where $a_{1}$ represents the first term and $a_{2}$ represents the second term.
Here, first term is $20$ and second term is $14$ .
Substitute $a_{1} = 20$ and $a_{2} = 14$ in the equation $d = a_{2} - a_{1}$ .
$d = 14 - 20 ∴ d = - 6$
Step-2 :Find the $12^{t h}$ term of the given sequence.
The formula to calculate the $n^{t h}$ term is $a_{n} = a_{1} + (n - 1) d$
where $a_{n}$ is the $n^{t h}$ term , $n$ is a positive integer and $d$ is the common difference.
Substitute $n = 12, a_{1} = 20, d = - 6$ in the equation $a_{n} = a_{1} + (n - 1) d$ .
$a_{12} = 20 + (12 - 1) (- 6) \Rightarrow a_{12} = 20 + 11 (- 6) \Rightarrow a_{12} = 20 - 66 ∴ a_{12} = - 46$
Hence, the $12^{t h}$ term of the sequence is $- 46$ .

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