To find $ a_{ 20 } $ we use formula

$$ \color{blue}{a_n = a_1 + (n-1)d}$$

In this example we have $ a_1 = \frac{ 1 }{ 3 },~~ d = \frac{ 1 }{ 8 } ~~,~~ n = 20 $. After substituting these values into the formula, we obtain:

$$ \begin{aligned} a_n &= a_1 + d(n-1) \\[1 em] a_{ 20 } &= \frac{ 1 }{ 3 } + (20-1) \cdot \frac{ 1 }{ 8 } \\[1 em] a_{ 20 } &= \frac{ 1 }{ 3 } + \frac{ 19 }{ 8 } \\[1 em] a_{ 20 } &= \frac{ 65 }{ 24 } \end{aligned} $$

The first few terms of this sequence are:

$$ \frac{ 1 }{ 3 }, ~~~\frac{ 11 }{ 24 }, ~~~\frac{ 7 }{ 12 }, ~~~\frac{ 17 }{ 24 }, ~~~\frac{ 5 }{ 6 } . . . $$

Arithmetic sequences