The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is

Solution(By Examveda Team)

$$\eqalign{ & {1^{st}}{\kern 1pt} {\text{Method}}: \cr & {1^{st}}{\kern 1pt} {\text{term}} = 5; \cr & {3^{rd}}{\kern 1pt} {\text{term}} = 15; \cr & {\text{Then}},{\kern 1pt} \,d = 5; \cr & {16^{th}}{\kern 1pt} {\text{term}} = a + 15d \cr & = 5 + 15 \times 5 = 80 \cr & {\text{Sum}} = {n \times \frac{{\left( {a + l} \right)}}{2}} \cr} $$
$$ = {{\text{no}}{\text{.}}{\kern 1pt} {\text{of}}{\kern 1pt} {\text{terms}} \times \frac{{ {{\text{first}}{\kern 1pt} {\text{term + last}}{\kern 1pt} {\text{term}}} }}{2}} $$
$$\eqalign{ & = {16 \times \frac{{\left( {5 + 80} \right)}}{2}} \cr & = 16 \times \frac{{85}}{2} \cr & = 8 \times 85 \cr & = 680 \cr} $$

2^nd Method(Thought Process):
Sum = number of terms × average of that AP
$$\eqalign{ & {\text{Sum}} = 16 \times {\frac{{\left( {5 + 80} \right)}}{2}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 16 \times \frac{{85}}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 8 \times 85 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 680 \cr} $$