Which of the following list of numbers forms an arithmetic progression?

(A) 1, -1, -3, -5, ........
(B) -1.2, -3.2, -5.2, -7.2, ........
(C) -2, 2, -2, 2, -2, ........
(D) $2,\frac{5}{2},3,\frac{7}{2},........$

Choose the correct answer from the options given below:

(A), (B) and (D) only

The correct answer is Option (1) → (A), (B) and (D) only

To determine which lists form an Arithmetic Progression (AP), we must check if there is a common difference ($d$) between consecutive terms. A sequence is an AP if $a_{n+1} - a_n = d$ (a constant).

Analysis of the Sequences:

(A) 1, -1, -3, -5, ...

• $-1 - 1 = -2$
• $-3 - (-1) = -2$
• $-5 - (-3) = -2$
• Result: Common difference $d = -2$. This is an AP.

(B) -1.2, -3.2, -5.2, -7.2, ...

• $-3.2 - (-1.2) = -2.0$
• $-5.2 - (-3.2) = -2.0$
• $-7.2 - (-5.2) = -2.0$
• Result: Common difference $d = -2.0$. This is an AP.

(C) -2, 2, -2, 2, -2, ...

• $2 - (-2) = 4$
• $-2 - 2 = -4$
• Result: The difference alternates between $4$ and $-4$. This is a geometric/oscillating sequence, not an AP.

(D) 2, $\frac{5}{2}$, 3, $\frac{7}{2}$, ...

(Note: Converting to decimals makes it easier: 2, 2.5, 3, 3.5, ...)

• $2.5 - 2 = 0.5$
• $3 - 2.5 = 0.5$
• $3.5 - 3 = 0.5$
• Result: Common difference $d = 0.5$ (or $\frac{1}{2}$). This is an AP.

Final Answer:

The sequences that form an arithmetic progression are (A), (B), and (D) only.