KA-0109Arithmetic sequences and the nth term
4 points, difficulty 2 of 3Level 3–4about 75sA staircase of blocks has 4 blocks in the first step, 7 in the second and 10 in the third. How many blocks are in the eighth step?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
How many steps of 3 to reach the eighth?
2The strategy
The blocks go up by the same amount each time. Getting from the first step to the eighth takes one fewer jump than eight, because the first step costs no jumps at all.
3The full solution
From step 1 to step 8 is 7 jumps of 3, so the eighth step is 4 + 7 x 3 = 25.
Solution
The reliable way
The counts rise by 3 each step. Getting from step 1 to step 8 takes 7 jumps, not 8, because step 1 is already there. So the eighth step has 4 + 7 x 3 = 4 + 21 = 25 blocks. Check the rule on a step you can see: step 3 should be 4 + 2 x 3 = 10, which matches. The transferable idea: count the gaps between terms, not the terms.
The elegant way
The nth step has 3n + 1 blocks, so the eighth has 24 + 1 = 25.
Why this is on the test: The one-fewer-jump rule is the defining off-by-one of the whole patterns strand.