In an A.P., the sum of three consecutive terms is $24$ and the sum of their squares is $194$. Find the numbers.

Question

Accepted Answer

Step 1: Let the numbers be $a - d$, $a$, $a + d$.
Step 2: $a - d + a + a + d = 24 \Rightarrow 3a = 24 \Rightarrow a = 8$
Step 3: $(a - d)^2 + a^2 + (a + d)^2 = 194 \Rightarrow (8 - d)^2 + 8^2 + (8 + d)^2 = 194$
Step 4: $\Rightarrow 192 + 2d^2 = 194 \Rightarrow d^2 = 1 \Rightarrow d = \pm 1$
Step 5: Numbers are $7, 8, 9$ or $9, 8, 7$

Question

Solution