a, A₁, A₂ ........... A_n, 100

Let d be the common difference of above A.P. then

$${{a + d} \over {100 - d}} = {1 \over 7}$$

$$ \Rightarrow 7a + 8d = 100$$ ...... (i)

and $$a + n = 33$$ ..... (ii)

and $$100 = a + (n + 1)d$$

$$ \Rightarrow 100 = a + (34 - a){{(100 - 7a)} \over 8}$$

$$ \Rightarrow 800 = 8a + 7{a^2} - 338a + 3400$$

$$ \Rightarrow 7{a^2} - 330a + 2600 = 0$$

$$ \Rightarrow a = 10,\,{{260} \over 7},$$ but $$a \ne {{260} \over 7}$$

$$\therefore$$ $$n = 23$$