Which of the following sequences are APs?
(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…………
(iii) and (iv)
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 a2–a1≠a3–a2, 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 a2–a1≠a3–a2, 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 a2–a1=a3−a2=a4−a3, 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 a2–a1=a3−a2=a4−a3, this sequence is an AP.