এটি সমান্তর ধারার অঙ্ক। এখানে A.P. হলো Arithmatic Progression যার অর্থ সমান্তর ধারা। প্রশ্নটিতে বলা হচ্ছে, একটি সমান্তর ধারার 3টি সংখ্যা x, y এবং z এর সমষ্টি 30 এবং সংখ্যাগুলোর বর্গের সমষ্টি 308 হলে সংখ্যা 3টি কত?

As the numbers are in Arithmatic Progression, their common difference will always be same. Here, we denote common difference by 'd'.

$$\begin{gathered} So, y-x=z-y \\ \Rightarrow y + y = x + z \\ \Rightarrow 2y = x + z.............. \left(i\right) \end{gathered}$$

Now, according to the first condition,

$$\begin{gathered} x + y + z = 30 \\ \Rightarrow y+x+z=30 \\ \Rightarrow y + 2y = 30 \\ \Rightarrow 3y = 30 \\ \Rightarrow y = \frac{30}{3} = 10 \\ \therefore y = 10 \end{gathered}$$

So, the second number = y = 10

As, common difference is d

∴ First number x = 10-d & Third number z = 10 + d

Now, according to the second condition $x^2+ y^2+z^2 = 308$

$$\begin{gathered} \Rightarrow {(10-d)}^2+{\left(10\right)}^2 + {(10+ d)}^2 = 308 \Rightarrow 100-20d + d^2 + 100+ 100+ 20d + d^2 = 308 \\ \Rightarrow 2d^2+300 = 308 \\ \Rightarrow 2d^2 = 308-300 = 8 \\ \Rightarrow d^2 = \frac{8}{2} = 4 \\ \Rightarrow d = \sqrt{4} = \pm 2 \end{gathered}$$

∴ The common difference will be '+2' Or '-2'

When the common difference d = 2

Then, the first number x = 10-2 = 8; Second number y = 10 & Third number z = 10+2 = 12

Again, when the common difference d = -2

Then, the first number x = 10-(-2)=10+2 = 12

Second number y = 10 & Third number z = 10+ (-2)=10-2=8

So, the numbers of the arithmatic progression may be 8, 10, 12 or 12, 10, 8