Dots are arranged in triangles as shown. Do the numbers of dots in Figures 1, 2, 3, 4, … form an arithmetic progression?

Class 10Arithmetic ProgressionsAlgebra — Sequences

Step-by-Step Solution

The figures have 1, 3, 6, 10 dots. Differences:

3 - 1 = 2, 6 - 3 = 3, 10 - 6 = 4

. A common difference is needed for an AP, so this is not an AP.

Answer: D) No, the differences 2, 3, 4 are not equal

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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