Question

Which of the following sequences are A.P.? If they are A.P. find the common difference.

i. 0.3, 0.33, 0.333,..

ii. 0, -4, -8, -12, ....

iii.-\(\frac{1}{5}\), -\(\frac{1}{5}\), -\(\frac{1}{5}\), ....

Answer 1

i. The given sequence is 0.3, 0.33, 0.333, … 

Here, t₁ = 0.3, t₂ = 0.33, t₃ = 0.333 

∴ t₂ -t₁ = 0.33 – 0.3 = 0.03 

t₃ – t₂ = 0.333 – 0.33 = 0.003 

∴ t₂ – t₁ ≠ t₃ – t₂

The difference between two consecutive terms is not constant. 

∴ The given sequence is not an A.P.

ii. The given sequence is 0, -4, -8, -12,… 

Here, t₁ = 0, t₂ = -4, t₃ = -8, t₄ = -12 

∴ t₂ – t₁ = -4 – 0 = -4 

t₃ – t₂ = -8 – (-4) = -8 + 4 = -4 

t₄ – t₃ = -12 – (-8) = -12 + 8 = -4 

∴ t₂ – t₁ = t₃ – t₂ = … = - 4 = d = constant 

The difference between two consecutive terms is constant. 

∴ The given sequence is an A.P. and common difference (d) = -4.

The difference between two consecutive terms is constant. 

∴ The given sequence is an A.P. and common difference (d) = 0.