Question

The sum of the first and third term of an arithmetic progression is 12 and the product of first and second term is 24, then the first term is?
(a) 1
(b) 8
(c) 4
(d) 6

Answer 1

Hint: Assume the first term of the AP as a and the common difference as d. Find the second and third term using these assumptions. Take the sum of the first and third term, equate it with 12 to form the first equation. Take the product of the first and second term, equate it with 24 to form the second equation. Substitute the value of (a + d) from equation (1) in (2) to get the answer.

Complete step by step solution:
Here we have been provided with an AP, and two relations related to its terms. We are asked to find the first term of the AP. Let us assume the first term of the AP is a and the common difference as d, so the three consecutive terms of the AP can be given as a, (a + d) and (a + 2d) respectively.
(i) It is given that the sum of the first and third term is equal to 12, so we have,
$\begin{align}
  & \Rightarrow a+\left( a+2d \right)=12 \\
 & \Rightarrow 2a+2d=12 \\
 & \Rightarrow a+d=6........\left( 1 \right) \\
\end{align}$
(ii) It is given that the product of the first and second term is equal to 24, so we have,
\[\Rightarrow a\times \left( a+d \right)=24........\left( 2 \right)\]
Substituting the value of (a + d) form equation (1) in equation (2) we get,
\[\begin{align}
  & \Rightarrow a\times 6=24 \\
 & \therefore a=4 \\
\end{align}\]
Therefore, the first term of the AP is equal to 4. Hence, option (c) is the correct answer.

Note: Do not assume the three terms of the AP as three different variables as it may create confusion. Note that sometimes we have to form a quadratic equation by the substitution of one variable in terms of the other to find the values of a or d. In such a case two values of a or d can be obtained where for the positive value of d the AP will be increasing and for the negative value of d the AP will be decreasing.