Question

Which of the following lists of numbers are Arithmetic Progressions? Justify.

i. 3, 6, 12, 24, … 

ii. 22, 26, 28, 31, … 

iii. 0.5, 2, 3.5, 5, …

Answer 1

i. The given list of numbers is 3, 6, 12, 24, …

Here, t₁ = 3, t₂ = 6, t₃ = 12, t₄ = 24

\(\therefore\) t₂ – t₁ = 6 – 3 = 3

t₃ – t₂ = 12 – 6 = 6

t₄ – t₃ = 24 – 12 = 12

\(\therefore\) t₂ – t₁ ≠ t₃ – t₂ ≠ t₄ – t₃

The difference between two consecutive terms is not constant.

\(\therefore\) The given list of numbers is not an A.P.

ii. The given list of numbers is 22, 26, 28, 31, …

Here, t₁ = 22, t₂ = 26, t₃ = 28, t₄ = 31

\(\therefore\) t₂ – t₁ = 26 – 22 = 4

t₃ – t₂ = 28 – 26 = 2

t₄ – t₃ = 31 – 28 = 3

\(\therefore\) t₂ – t₁ ≠ t₃ – t₂ ≠ t₄ – t₃

The difference between two consecutive terms is not constant.

\(\therefore\) The given list of numbers is not an A.P.

iii. The given list of numbers is 0.5, 2, 3.5, 5, …

Here, t₁ = 0.5, t₂ = 2, t₃ = 3.5, t₄ = 5

\(\therefore\) t₂ – t₁ = 2 – 0.5 = 1.5

t₃ – t₂ = 3.5 – 2 = 1.5

t₄ – t₃ = 5 – 3.5 = 1.5

\(\therefore\) t₂ – t₁ = t₃ – t₂ = ... = 1.5 = constant

The difference between two consecutive terms is constant.

\(\therefore\) The given list of numbers is an A.P.