KA-0171Recursive rules
3 points, difficulty 1 of 3Level 11–12about 75sA sequence is defined by a(1) = 2 and a(n+1) = 2 a(n) - 1 for every n. What is a(6)?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Just generate the terms one at a time; six is not many.
2The strategy
a1 = 2, a2 = 2(2) - 1 = 3, a3 = 2(3) - 1 = 5. Keep going.
3The full solution
a4 = 9, a5 = 17, a6 = 33.
Solution
The reliable way
Generate the terms directly: a1 = 2; a2 = 2 x 2 - 1 = 3; a3 = 2 x 3 - 1 = 5; a4 = 2 x 5 - 1 = 9; a5 = 2 x 9 - 1 = 17; a6 = 2 x 17 - 1 = 33. With only six terms wanted there is no reason to look for a closed form, and doing so is where mistakes creep in. If you do want one, notice each term is one more than a power of two: a(n) = 2^(n-1) + 1, and 2^5 + 1 = 33. The transferable idea: when only a few terms are wanted, generating them beats deriving a formula.
The elegant way
Each term is one more than a power of 2, so a(6) = 2^5 + 1 = 33.
Why this is on the test: Recursive rules reward just computing, and the marks are lost by reaching for a formula or stopping one term short.