Question

Whether following sequences are APs or not?

(i) 2, 4, 8, 16……..

(ii) 2, 3, 5, 7, 11……..

(iii) -1, -1.25, -1.5, -1.75…….

(iv) 1, -1, -3, -5, -7…………

Answer 1

Consider each of the sequences.
(i) 2, 4, 8, 16……..

Difference between the first two terms = 4 - 2 = 2

Difference between the third and second terms = 8 - 4 = 4

Since $a_2–a_1\ne a_3–a_2$, this sequence is not an AP.

(ii) 2, 3, 5, 7, 11……..

Difference of the first two terms = 3 - 2 = 1

Difference of the third and second terms = 5 - 3 = 2

Since $a_2–a_1\ne a_3–a_2$, this sequence is not an AP.

(iii) -1, -1.25, -1.5, -1.75…….

Difference of the first two terms = -1.25 – (-1) = -0.25

Difference of the third and second terms = -1.5 – (-1.25) = -0.25

Difference of the fourth and third terms = -1.75 – (-1.5) = -0.25

Since $a_2–a_1=a_3-a_2=a_4-a_3$, this sequence is an AP.

(iv) 1, -1, -3, -5, -7…………

Difference of the first two terms = -1 – 1 = -2

Difference of the third and second terms = -3 – (-1) = -2

Difference of the fourth and third terms = -5 – (-3) = -2

Since $a_2–a_1=a_3-a_2=a_4-a_3$, this sequence is an AP.