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(A) Let the first terms of the two arithmetic progressions be $a_1$ and $a_2$,and their common differences be $d_1$ and $d_2$ respectively.Given the r

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$53 : 155$

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(A) Let the first terms of the two arithmetic progressions be $a_1$ and $a_2$,and their common differences be $d_1$ and $d_2$ respectively.Given the ratio of the sum of $n$ terms: $\frac{S_n}{S'_n} = \frac{2n + 3}{6n + 5}$.We know that the $m^{th}$ term $t_m = a + (m-1)d$.The ratio of the $m^{th}$ terms is $\frac{t_m}{t'_m} = \frac{a_1 + (m-1)d_1}{a_2 + (m-1)d_2}$.To find the $13^{th}$ term,we set $m-1 = 12$,so $m = 13$.We substitute $n = 2m - 1 = 2(13) - 1 = 25$ into the sum ratio formula.$\frac{t_{13}}{t'_{13}} = \frac{a_1 + 12d_1}{a_2 + 12d_2} = \frac{2(25) + 3}{6(25) + 5}$.$\frac{t_{13}}{t'_{13}} = \frac{50 + 3}{150 + 5} = \frac{53}{155}$.

If the ratio of the sum of $n$ terms of two arithmetic progressions is $2n + 3 : 6n + 5$,then the ratio of their $13^{th}$ terms is.......

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