KA-0103Recursive rules
4 points, difficulty 2 of 3Level 5–6about 75sA sequence starts at 3. Each term after that is double the one before it, minus 1. What is the sixth term?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Write the terms out one at a time.
2The strategy
A recursive rule tells you how to get the next term from the last one. There is no formula to jump to - generate the terms in order and count them as you go.
3The full solution
3, then 5, then 9, then 17, then 33, then 65. Counting the starting 3 as the first term, the sixth is 65.
Solution
The reliable way
Apply the rule term by term, writing each one down and numbering it. Term 1 is 3. Term 2 is 2 x 3 - 1 = 5. Then 2 x 5 - 1 = 9, then 17, then 33, then 65. Counting carefully, the sixth term is 65. The transferable idea: with a recursive rule, number the terms as you write them - almost every error here is being one term out.
The elegant way
Each term is one more than a power of two: 3, 5, 9, 17, 33, 65 are 2 to the powers 1 to 6, each plus one.
Why this is on the test: Recursive rules cannot be jumped, only generated, and being one term out is by far the most common way to lose the mark.