પ્રત્યેક પ્રાકૃતિક સંખ્યા $n$ માટે બે સમાંતર શ્રેણીઓનાં પ્રથમ $n$ પદોના સરવાળાનો ગુણોત્તર $(3 n+8):(7 n+15)$ હોય, તો તેમનાં $12$ માં પદનો ગુણોત્તર શોધો. 

Let $a_{1}, a_{2}$ and $d_{1}, d_{2}$ be the first terms and common difference of the first and second arithmetic progression, respectively. According to the given condition, we have

$\frac{{{\rm{ Sum}}\,\,{\rm{to}}\,\,{\rm{nterms}}\,\,{\rm{of}}\,\,{\rm{ first }}\,{\rm{A}}{\rm{.P}}{\rm{. }}}}{{{\rm{ Sumt}}\,\,\,{\rm{on}}\,\,{\rm{ terms }}\,\,{\rm{of }}\,\,{\rm{second }}\,\,{\rm{A}}{\rm{.P}}{\rm{. }}}} = \frac{{3n + 8}}{{7n + 15}}$

or    $\frac{\frac{n}{2}\left[2 a_{1}+(n-1) d_{1}\right]}{\frac{n}{2}\left[2 a_{2}+(n-1) d_{2}\right]}=\frac{3 n+8}{7 n+15}$

or    $\frac{2 a_{1}+(n-1) d_{1}}{2 a_{2}+(n-1) d_{2}}=\frac{3 n+8}{7 n+15}$       .........$(1)$

Now    $\frac{{{{12}^{{\rm{th }}}}{\rm{ term}}\,\,{\rm{of }}\,\,{\rm{first \,A}}{\rm{.P}}{\rm{. }}}}{{{{12}^{{\rm{th }}}}{\rm{ term}}\,\,{\rm{of }}\,\,{\rm{second \,A}}{\rm{.P }}}} = \frac{{{a_1} + 11{d_1}}}{{{a_2} + 11{d_2}}}$

$\frac{2 a_{1}+22 d_{1}}{2 a_{2}+22 d_{2}}=\frac{3 \times 23+8}{7 \times 23+15}$         [ By putting $n=23$ in $(1)$ ]

Therefore   $\frac{a_{1}+11 d_{1}}{a_{2}+11 d_{2}}=\frac{12^{\text {th }} \text { term of first A.P. }}{12^{\text {th }} \text { term of second A.P. }}=\frac{7}{16}$

Hence, the required ratio is $7: 16$