Correct option is (D) 1 + 2 + 3 + …………

(A) Given sequence is 1, 8, 27, .....

\(\because\) \(a_2-a_1\) = 8 - 1 = 7,

\(a_3-a_2\) = 27- 8 = 19

\(\because\) \(a_2-a_1\) \(\neq a_3-a_2\)

\(\therefore\) Given sequence is not an arithmetic progression.

(B) Given sequence is 1, 4, 9, .......

\(\therefore\) \(a_2-a_1\) = 4 - 1 = 3

& \(a_3-a_2\) = 9 - 4 = 5

\(\because\) \(3\neq5\)

\(\therefore\) \(a_2-a_1\) \(\neq a_3-a_2\)

\(\therefore\) Given sequence is not an arithmetic progression.

(C) Given sequence is 1, -2, 3, -4, .......

\(\therefore\) \(a_2-a_1\) = -2 - 1 = -3,

\(a_3-a_2\) = 3 - (-2)

= 3+2 = 5

\(\because\) \(-3\neq5\)

\(\therefore\) \(a_2-a_1\) \(\neq a_3-a_2\)

\(\therefore\) Given sequence is not an arithmetic progression.

(D) Given sequence is 1, 2, 3, ......

\(\therefore\) \(a_2-a_1\) = 2 - 1 = 1,

\(a_3-a_2\) = 3 - 2 = 1

\(\because\) \(a_2-a_1\) \(=a_3-a_2\)

\(\therefore\) Given sequence is an arithmetic progression (A.P.).

Correct option is D) 1 + 2 + 3 + …………