KA-0160Arithmetic sequences and the nth term
3 points, difficulty 1 of 3Level 9–10about 75sA sequence begins 7, 11, 15, 19, and continues with the same constant step. What is the 30th term?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the gaps between terms, not the terms themselves.
2The strategy
Going from the 1st term to the 30th takes 29 steps of 4.
3The full solution
7 + 29 x 4 = 7 + 116 = 123.
Solution
The reliable way
The step is 11 - 7 = 4. Reaching the 30th term from the 1st takes 29 steps, not 30, because the first term is already there before any step is taken. So the 30th term is 7 + 29 x 4 = 7 + 116 = 123. Check on a small case: the 2nd term should be 7 + 1 x 4 = 11, which matches. The transferable idea: the nth term is first + (n - 1) x step, and the minus one is where the marks go.
The elegant way
Every term is 3 less than a multiple of 4, so the 30th is 4 x 31 - 1 = 123.
Why this is on the test: The off-by-one between terms and gaps is the single most common arithmetic-sequence error.