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Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $\mathrm{S}_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7$, then $S_{15}-S_5$ is equal to:
$800$
$890$
$790$
$690$
Solution
$ \mathrm{S}_{10}=390 $
$ \frac{10}{2}[2 \mathrm{a}+(10-1) \mathrm{d}]=390 $
$ \Rightarrow 2 \mathrm{a}+9 \mathrm{~d}=78 $ $……(1)$
$ \frac{\mathrm{t}_{10}}{\mathrm{t}_5}=\frac{15}{7} \Rightarrow \frac{\mathrm{a}+9 \mathrm{~d}}{\mathrm{a}+4 \mathrm{~d}}=\frac{15}{7} \Rightarrow 8 \mathrm{a}=3 \mathrm{~d} $ $……(2)$
$ \text { From }(1) \&(2) \quad \mathrm{a}=3 \& \mathrm{~d}=8 $
$ \mathrm{~S}_{15}-\mathrm{S}_5=\frac{15}{2}(6+14 \times 8)-\frac{5}{2}(6+4 \times 8) $
$ =\frac{15 \times 118-5 \times 38}{2}=790$