If S₁ is sum of an arithmetic progression of ‘n’ odd number of terms and S₂ is sum of terms of series in odd places, n $$\frac{{{S_1}}}{{{S_2}}}$$

Odd numbers are 1, 3, 5, 7, 9, 11, 13, ...... n
∴ S₁ = Sum of odd numbers = n²
S₂ = Sum of number at odd places
3, 7, 11, 15, ......
a = 3, d = 7 - 3 = 4 and number of term = $$\frac{n}{2}$$
$$\eqalign{ & {S_2} = \frac{n}{{2 \times 2}}\left[ {2 \times 3 + \left( {\frac{n}{2} - 1} \right) \times 4} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{n}{4}\left[ {6 + 2n - 4} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{n}{4}\left[ {2n + 2} \right] \cr & \,\,\,\,\,\,\,\,\, = \frac{{n\left( {n + 1} \right)}}{2} \cr & \therefore \frac{{{s_1}}}{{{s_2}}} = \frac{{{n^2} \times 2}}{{n\left( {n + 1} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2n}}{{n + 1}} \cr} $$