Question: The start of a sequence is shown below.
To work out a term of the sequence, the previous two terms are added together.
Work out the three missing terms in the sequence.
3, __, 7, __, __, 29, ...
1. **State the problem:** We have a sequence where each term is the sum of the previous two terms. The sequence starts as $3, \_, 7, \_, \_, 29, ...$ and we need to find the three missing terms.
2. **Understand the rule:** If we denote the terms as $a_1, a_2, a_3, a_4, a_5, a_6, ...$, then the rule is:
$$a_n = a_{n-1} + a_{n-2}$$
3. **Assign known values:**
$$a_1 = 3, \quad a_3 = 7, \quad a_6 = 29$$
4. **Find $a_2$:** Using the rule for $a_3$:
$$a_3 = a_2 + a_1$$
$$7 = a_2 + 3$$
$$a_2 = 7 - 3 = 4$$
5. **Find $a_4$:** Using the rule for $a_4$:
$$a_4 = a_3 + a_2$$
$$a_4 = 7 + 4 = 11$$
6. **Find $a_5$:** Using the rule for $a_5$:
$$a_5 = a_4 + a_3$$
$$a_5 = 11 + 7 = 18$$
7. **Check $a_6$:** Using the rule for $a_6$:
$$a_6 = a_5 + a_4$$
$$a_6 = 18 + 11 = 29$$
This matches the given term, confirming our calculations.
**Final answer:** The missing terms are $4, 11, 18$.
Thus, the full sequence starts as:
$$3, 4, 7, 11, 18, 29, ...$$