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This can be determined by the arithmetic sequence formula . Hence ,x+6 is the general equation for find the [tex]\bold{n^{th}}[/tex] term in the sequence i.e. [tex]\bold{a_n}[/tex].
In this question given that, a sequence i.e.
[tex]a_3[/tex]= 0, [tex]a_4[/tex] = 6, [tex]a_5[/tex] = 12, [tex]a_6[/tex] = 18, and [tex]a_7[/tex] = 24
We have to find the [tex]\bold{n^{th}}[/tex] term in the sequence i.e. [tex]\bold{a_n}[/tex].
From the sequence, now find common difference is defined by formula
i.e. [tex]d= a_n_+_1-a_n[/tex]
Therefore,
[tex]\begin{aligned}d_1&= a_4-a_3 \\&=6-0 \\&=6\end{aligned}[/tex] [tex]\begin{aligned}d_2&= a_5-a_4 \\&=12-6 \\&=6\end{aligned}[/tex] [tex]\begin{aligned}d_3&= a_6-a_5 \\&=18-12 \\&=6\end{aligned}[/tex] [tex]\begin{aligned}d_4&= a_7-a_6 \\&=24-18 \\&=6\end{aligned}[/tex]
So, the difference is remains constant with the common difference 6.
We know that, the formula of arithmetic sequence is
[tex]\bold{a_n=a+(n-1)d}[/tex]
Therefore, from the above it can be determined that
[tex]a_1 =-12 \:and \:a_2=-6[/tex]
Therefore, the [tex]\bold{n^{th}}[/tex] term of the sequence is
[tex]\begin{aligned}\bold{a_n&=-12+(n-1)6\\&=-12+6n-6\\a_n&=6n-18}\end{aligned}[/tex]
For example, we have to find [tex]\bold{8^{th}}[/tex] term ,
[tex]\begin{aligned}a_8&=-12+(8-1)6\\&=-12+(7)6\\&=30\end{aligned}[/tex]
Hence , from the above solution we find that x+6 is the general equation for find the[tex]n^{th}[/tex] term in the sequence i.e. [tex]a_n[/tex].
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