Question

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

i. 4, 3, 2, 1, … 

ii. -10, -13, -16, -19, …

Answer 1

i. The given list of numbers is 4, 3, 2, 1, …

Here, t₁ = 4, t₂ = 3, t₃ = 2, t₄ = 1

\(\therefore\) t₂ – t₁ = 3 – 4 = –1

t₃ – t₂ = 2 – 3 = –1

t₄ – t₃ = 1 – 2 = –1

\(\therefore\) t₂ – t₁ = t₃ – t₂ = ... = –1 = constant

The difference between two consecutive terms is constant.

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

ii. The given list of numbers is –10, –13, –16, –19, …

Here, t₁ = –10, t₂ = –13, t₃ = –16, t₄ = –19

\(\therefore\) t₂ – t1 = – 13 – (–10) = –3

t₃ – t₂ = – 16 – (–13) = – 3

t₄ – t₃ = –19 – (–16) = –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.