Q.The Fibonacci sequence is defined by $1 = a_1 = a_2$ and $a_n = a_{n-1} + a_{n-2}$, $n > 2$. Find $\dfrac{a_{n+1}}{a_n}$, for $n = 1, 2, 3, 4, 5$.

Concept: Sequence Terms Evaluation — we compute the first few terms of the Fibonacci sequence and then form the required ratios.

Step 1: Write the given terms.

$a_1 = 1$, $a_2 = 1$.

Step 2: Generate the next terms using $a_n = a_{n-1} + a_{n-2}$.

$a_3 = 1+1 = 2$, $a_4 = 2+1 = 3$, $a_5 = 3+2 = 5$, $a_6 = 5+3 = 8$.

Step 3: Compute $\dfrac{a_{n+1}}{a_n}$ for $n = 1,2,3,4,5$.

• $n=1$: $\dfrac{a_2}{a_1} = \dfrac{1}{1} = 1$
• $n=2$: $\dfrac{a_3}{a_2} = \dfrac{2}{1} = 2$ …

The key insight here is that we are not being asked for a formula or a limit — just the exact rational values for the first five terms of the ratio sequence. The Fibonacci numbers themselves grow quickly, but the ratios settle down. Let’s build them step by step.

1. Write down the first few Fibonacci terms.

   The definition gives $a_1 = 1$, $a_2 = 1$. Then:

   • $a_3 = a_2 + a_1 = 1 + 1 = 2$
   • $a_4 = a_3 + a_2 = 2 + 1 = 3$
   • $a_5 = a_4 + a_3 = 3 + 2 = 5$
   • $a_6 = a_5 + a_4 = 5 + 3 = 8$

   So the sequence begins: $1, 1, 2, 3, 5, 8, \dots$

2. Now form each ratio $\frac{a_{n+1}}{a_n}$ for $n = 1, 2, 3, 4, 5$.

   For $n=1$: $\frac{a_2}{a_1} = \frac{1}{1} = 1$

   For $n=2$: $\frac{a_3}{a_2} = \frac{2}{1} = 2$

   For $n=3$: $\frac{a_4}{a_3} = \frac{3}{2} = 1.5$

   For $n=4$: $\frac{a_5}{a_4} = \frac{5}{3} \approx 1.666\ldots$ (exactly $1\frac{2}{3}$)

   For $n=5$: $\frac{a_6}{a_5} = \frac{8}{5} = 1.6$

3. Notice the pattern. …