Question

Let {an}n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

• A. If \left\\{\frac{a_{n+1}}{a_n} \right\\}_{𝑛≥1} is a convergent sequence, then {an}n≥1 is also a convergent sequence
• B. If {an}n≥1 is a bounded sequence, then {an}n≥1 is a convergent sequence
• C. If |an+2 - an+1| ≤ $\frac{3}{4}$ |an+1 - an| for all n ≥ 1, then {an}n≥1 is a Cauchy sequence
• D. If {an}n≥1 is a Cauchy sequence, then {an}n≥1 is also a Cauchy sequence

Answer 1

Answer: (C)

If |an+2 - an+1| ≤ $\frac{3}{4}$ |an+1 - an| for all n ≥ 1, then {an}n≥1 is a Cauchy sequence

Answer 2

The correct option is (C) : If |an+2 - an+1| ≤ $\frac{3}{4}$ |an+1 - an| for all n ≥ 1, then {an}n≥1 is a Cauchy sequence.