Find the next term: 1, 1, 2, 3, 5, 8, ...

Question

Find the next term in the sequence: 1, 1, 2, 3, 5, 8, …

Also explain the pattern that governs this sequence.

Solution — Step by Step

List the terms with their positions:

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

Check if each term is the sum of the two before it:

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

The rule is clear: each term equals the sum of the two immediately preceding terms. This is the defining rule of the Fibonacci sequence, named after the Italian mathematician Leonardo of Pisa (Fibonacci), who introduced it to Europe in 1202.

The recurrence relation is: $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$ , with $a_1 = 1$ and $a_2 = 1$ .

The next term is $a_7$ :

a_7 = a_5 + a_6 = 5 + 8 = \mathbf{13}

The sequence continues: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Why This Works

The Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ is one of the simplest second-order recurrence relations. The reason this specific pattern is so famous is that it shows up naturally in biological growth — the spiral arrangement of seeds in a sunflower, the branching of trees, the spiral of a nautilus shell, and the arrangement of leaves on a stem all follow Fibonacci numbers.

In mathematics, the ratio of consecutive Fibonacci numbers $a_{n+1}/a_n$ approaches the Golden Ratio $\phi = (1 + \sqrt{5})/2 \approx 1.618$ as $n$ increases. For example, $8/5 = 1.6$ and $13/8 = 1.625$ — both close to $\phi$ .

Alternative Method

We can verify using the closed-form formula (Binet’s formula), though this is beyond school level:

F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]

For school exams, simply applying the recurrence $a_n = a_{n-1} + a_{n-2}$ is sufficient and expected.

For pattern recognition questions in competitive exams, always check three operations: (1) constant difference (arithmetic), (2) constant ratio (geometric), and (3) each term = sum/product of previous terms. If none work, look for alternating patterns or squares. The Fibonacci rule — each term = sum of two preceding — is specifically tested in olympiad papers and aptitude tests.

Common Mistake

Some students see the pattern 1, 1, 2, 3, 5, 8 and think it’s arithmetic or geometric because the early terms look deceivingly simple. Check: $2-1=1$ , $3-2=1$ , $5-3=2$ , $8-5=3$ — the differences are themselves 1, 1, 2, 3… which is the Fibonacci pattern again! So this is NOT arithmetic (constant differences) and NOT geometric (constant ratio). It’s a recurrence — each term depends on the two before it.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next