Try this beautiful problem from IIT JAM 2018 which requires knowledge of Real Analysis (Limit of a Sequence).
Let $a_n=\frac{b_{n+1}}{b_n}$ where $b_1=1, b_2=1$ and $b_{n+2}=b_n+b_{n+1}$ , Then $\lim\limits_{n \to \infty} a_n$ is
Real Analysis
Sequence of Reals
Limit of a Sequence
Answer: $\frac{1+\sqrt5}{2}$
IIT JAM 2018 (Problem 2)
Advanced Calculus by Patrick Fitzpatrick
Given that, $a_n=\frac{b_{n+1}}{b_n}$
$\Rightarrow \lim\limits_{n \to \infty} a_n = \lim\limits_{n \to \infty} \frac{b_{n+1}}{b_n}= \mathcal{L} $ (say)
Now we know that , $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} a_{n+1} $
$\Rightarrow \mathcal{L}=\lim\limits_{n \to \infty} a_{n+1}$
Can you find an equation on $\mathcal{L}$ from which the value of $\mathcal{L}$ can be obtained.
$\mathcal{L}= \lim\limits_{n \to \infty } a_{n+1}$
$= \lim\limits_{n \to \infty} \frac{b_{n+2}}{b_{n+2}}$
$=\lim\limits_{n\to \infty} \frac{b_{n+1}+b_n}{b_{n+1}}$ [By the given recurrence relation]
$=\lim\limits_{n\to \infty} \left(1+\frac{b_n}{b_{n+1}}\right)$
$=1+\lim\limits_{n \to \infty} \frac{b_n}{b_{n+1}}$
$=1+\frac{1}{\lim\limits_{n\to\infty}\frac{b_{n+1}}{b_n}}$
$=1+\frac{1}{\mathcal{L}}$
Now the value of $\mathcal{L}$ can be easily obtained
i.e., $\mathcal{L}=1+\frac{1}{\mathcal{L}}$
$\Rightarrow \mathcal{L}^2-\mathcal{L}-1=0$
$\Rightarrow \mathcal{L}=\frac{1\pm \sqrt{5}}{2}$
$\Rightarrow \mathcal{L}=\frac{1+\sqrt{5}}{2}$ [Since $a_n>0$] [ANS]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
