Find the ratio of twelfth terms if the sum of first n terms of two AP's are $3n+8,7n+15$

Question

The sum of first n terms of two AP's are $3n+8,7n+15$. Then the ratio of their twelfth term is:

$$ \frac{a_{12}}{A_{12}}=\frac{a+11d}{A+11D}=\frac{2a+22d}{2A+22D}=\frac{s_{23}}{S_{23}}=\frac{3(23)+8}{7(23)+15}=\frac{77}{176}=\frac{7}{16} $$ Fine but if I do the following $$ a_{12}=s_{12}-s_{11}=44-41 =3\\ A_{12}=S_{12}-S_{11}=99-92=7\\ \frac{a_{12}}{A_{12}}=\frac{3}{7} $$ Why am I getting diferent results for method 1 and 2 ?. Is the question wrong because the sum of terms of AP is a quadratic expression in $n$ ?

Note: This was asked as a multiple choice question with options including both the results of method 1 and 2.

Answer 1

The question should have been

The ratio of the sum of first $n$ terms of two AP's are $3n+8:7n+15$

instead of

The sum of first $n$ terms of two AP's are $3n+8,7n+15$

Answer 2

Yes the question is wrong. Either it is not an A.P. or the sum of n terms can't be linear(it can be of the form $S_n=an$ if the common difference is zero).