Solve the equation 2 + 5 + 8 + …. + x = 155.

The above series is an A.P. with a = 2, d = 5 - 2 = 3 and Sum = 155.

Let x be nth term so, a_n = a + (n - 1)d.

⇒ x = 2 + (n - 1)3
⇒ x = 2 + 3n - 3
⇒ x = 3n - 1.

Sum = $\dfrac{n}{2}[a + l]$

$\therefore 155 = \dfrac{n}{2}[2 + x] \\[1em] \Rightarrow \dfrac{n}{2}[2 + 3n - 1] = 155 \\[1em] \Rightarrow \dfrac{n}{2}[3n + 1] = 155 \\[1em] \Rightarrow 3n^2 + n = 155 \times 2 \\[1em] \Rightarrow 3n^2 + n - 310 = 0 \\[1em] \Rightarrow 3n^2 - 30n + 31n - 310 = 0 \\[1em] \Rightarrow 3n(n - 10) + 31(n - 10) = 0 \\[1em] \Rightarrow (3n + 31)(n - 10) = 0 \\[1em] \Rightarrow 3n + 31 = 0 \text{ or } n - 10 = 0 \\[1em] \Rightarrow n = -\dfrac{31}{3} \text{ or } n = 10. \\[1em]$

Since, number of terms cannot be in fraction so, $n$ ≠ $-\dfrac{31}{3}.$

∴ x = 3n - 1 = 3 × 10 - 1 = 30 - 1 = 29.

Hence, the value of x is 29.