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 |a_n+2 - a_n+1| ≤ \(\frac{3}{4}\) |a_n+1 - a_n| for all n ≥ 1, then {a_n}_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

The Correct Option is C

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