Let S _ n denote the sum of the first n terms of an arithmetic...

Given: $\dfrac{{a}_{10}}{{a}_{5}} = \dfrac{15}{7}$

$⇒ \dfrac{a + 9d}{a + 4d} = \dfrac{15}{7}$

$⇒ 7a + 63d = 15a + 60d$

$⇒ 3d = 8a$

Also, ${S}_{10} = 390$

$⇒ \dfrac{10}{2}\left[2a + 9d\right] = 390$

$⇒ 2a + 24a = 78$

$⇒ a = 3$

$⇒ d = 8$

$⇒ {S}_{15} - {S}_{5} = \dfrac{15}{2}\left(6 + 14 \times 8\right) - \dfrac{5}{2}\left(6 + 4 \times 8\right)$

$⇒ {S}_{15} - {S}_{5} = \dfrac{15}{2} \times 118 - \dfrac{5}{2} \times 38$

$⇒ {S}_{15} - {S}_{5} = 885 - 95$

$⇒ {S}_{15} - {S}_{5} = 790$