Which one of these sequences is NOT an arithmetic progression?

Class 10Arithmetic ProgressionsAlgebra — Sequences

A-7, -4, -1, 2, …
B12, 9, 6, 3, …
C2, 8, 18, 32, …
D0.5, 1, 1.5, 2, …

Step-by-Step Solution

Check the successive differences. -7, -4, -1, 2 gives 3, 3, 3 (AP). 12, 9, 6, 3 gives -3, -3, -3 (AP). 0.5, 1, 1.5, 2 gives 0.5 every time (AP). But 2, 8, 18, 32 gives 6, 10, 14 — the difference keeps growing, so it is not an AP.

Answer: C) 2, 8, 18, 32, …

Key idea: Inserting arithmetic means

Between two numbers a and

b

, you can insert

k

'arithmetic means' so that the resulting sequence is in AP.

**Setup**: a,

m

₁,

m

₂, …,

m_{k}, b

. Total of

k + 2

terms in AP, so:
- First term =

a

, last term =

b

.
- Common difference

d = (b - a) / (k + 1)

(because there are

k + 1

'gaps').

**Example**: insert 3 arithmetic means between 4 and 16.

d = (16 - 4) / (3 + 1) = 3

. Means: 7, 10, 13. Check: 4, 7, 10, 13, 16 —

d = 3

✓.

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