Question

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

i. 1, 3, 6, 10, …

ii. 3, 5, 7, 9, 11, 

iii. 1, 4, 7, 10, … 

Answer 1

i. The given list of numbers is 1, 3, 6, 10, …

Here, t₁ = 1, t₂ = 3, t₃ = 6, t₄ = 10

\(\therefore\) t₂ – t₁ = 3 – 1 = 2

t₃ – t₂ = 6 – 3 = 3

t₄ – t₃ = 10 – 6 = 4

\(\therefore\) 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 3, 5, 7, 9, 11, ...

Here, t₁ = 3, t₂ = 5, t₃ = 7, t₄ = 9, t₅ = 11

\(\therefore\) t₂ – t1 = 5 – 3 = 

 t₃ – t₂ = 7 – 5 = 2

t₄ – t₃ = 9 – 7 = 2

t₅ – t₄ = 11 – 9 = 2

\(\therefore\) t₂ – t₁ = t₃ – t₂ = ... = 2 = constant

The difference between two consecutive terms is constant.

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

iii. The given list of numbers is 1, 4, 7, 10, …

Here, t₁ = 1, t₂ = 4, t₃ = 7, t₄ = 10

\(\therefore\) t₂ – t₁ = 4 – 1 = 3

t₃ – t₂ = 7 – 4 = 3

t₄ – t₃ = 10 – 7 = 3

\(\therefore\) t₂ – t₁ = t₃ – t₂ = ... = 3 = constant

The difference between two consecutive terms is constant.

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