If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is :

Answer: Option C

Let three consecutive terms of an increasing A.P. be a - d, a + d where a is the first term and d be the common difference
$$\begin{array}{rl}& \therefore a-d+a+a+d=51\\ & \Rightarrow 3a+51\\ & \therefore a=\frac{51}{3}=17\end{array}$$
and product of the first and third terms
$$\begin{array}{rl}& =\left(a-d\right)\left(a+d\right)=273\\ & \Rightarrow a^2-d^2=273\\ & \Rightarrow {\left(17\right)}^2-d^2=273\\ & \Rightarrow 289-d^2=273\\ & \Rightarrow d^2=289-273\\ & \Rightarrow d^2=16\\ & \Rightarrow d^2={\left(\pm 4\right)}^2\\ & \therefore d=\pm 4\\ & \because \text{The A}\text{.P}\text{.}\text{is}\text{increasing}\\ & \therefore d=4\\ & \text{Now}\text{third}\text{term}=a+d\\ & =17+4=21\end{array}$$