Fibonacci Sequence and Fibonacci-Type Sequences

In a Fibonacci-type sequence, each term is the sum of the two terms before it.

Add the last two terms: $14+23=37$.

So the next term is 37.

Check the rule on each option: every term should be the sum of the two before it.

Here: $3+7=10$, $7+10=17$ and $10+17=27$.

The rule holds all the way through, so this is a Fibonacci-type sequence.

The other options either grow by adding or multiplying by a fixed amount, or only fit the rule for the first pair.

Each term is the sum of the two before it, so the 3rd term equals the 1st term plus the missing 2nd term.

This gives $6+\text{missing term}=15$, so the missing term is $15-6=9$.

Each term is the sum of the two before it, so keep adding the last two terms.

$4+6=10$, then $6+10=16$, then $10+16=26$, then $16+26=42$.

The terms are $4,6,10,16,26,42$, so the 6th term is 42.

We know the 4th and 5th terms, so the sequence so far is $\_,\_,\_,25,40,\dots$ and we need to fill in the earlier terms.

The 5th term is the 3rd term plus the 4th term, so the 3rd term is $40-25=15$.

The 4th term is the 2nd term plus the 3rd term, so the 2nd term is $25-15=10$.

So the 2nd term is 10.

Each term is the sum of the two before it, so work backwards by subtracting.

4th term: $51-31=20$.

3rd term: $31-20=11$.

2nd term: $20-11=9$.

1st term: $11-9=2$.

So the sequence is $2,9,11,20,31,51$ and the 1st term is 2.