KA-0030Arithmetic sequences and the nth term
4 points, difficulty 2 of 3Level 5–6about 75sA sequence starts 4, 7, 10, 13 and continues in the same way. What is the 20th term?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
How many steps from the first term to the twentieth?
2The strategy
The terms go up by a fixed amount. To reach the 20th term from the 1st you take one fewer step than 20, because the first term costs no steps at all.
3The full solution
The step is 3. From term 1 to term 20 is 19 steps, so the 20th term is 4 + 19 x 3 = 4 + 57 = 61.
Solution
The reliable way
Each term is 3 more than the one before. Getting from the 1st term to the 20th takes 19 steps, not 20, because the first term is already there. So the 20th term is 4 + 19 x 3 = 4 + 57 = 61. Check on a small case: the 4th term should be 4 + 3 x 3 = 13, which matches. The transferable idea: count the gaps, not the terms.
The elegant way
The nth term is 3n + 1, so the 20th is 60 + 1 = 61.
Why this is on the test: The one-fewer-step rule is the single most common off-by-one in the patterns strand, and every distractor here is a version of getting it wrong.