The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19. Find the first term?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Sum of A}}{\text{.P}}{\text{.}} = \frac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] \cr & {n^{{\text{th}}}}{\text{term}} = a + \left( {n - 1} \right)d \cr & {3^{{\text{rd}}}}{\text{term}} = a + \left( {3 - 1} \right)d = 19 \cr & a + 2d = 19\,......\,\left( {\text{i}} \right) \cr & {\text{Sum of 10 term}} = \frac{{10}}{2}\left[ {2a + 9d} \right] = 390 \cr & 2a + 9d = 78\,......\,\left( {{\text{ii}}} \right) \cr & a + 2d = 19\,......\,\left( {\text{i}} \right) \cr & {\text{From equation }}\left( {\text{i}} \right){\text{ and }}\left( {{\text{ii}}} \right) \cr & \boxed{a = 3}\,\,\boxed{d = 8} \cr} $$