AMC 10 · 2014 · #7
Grade 11 algebrapatternPick an answer.
The three terms look unrelated only because they wear different radical signs. They are all roots of the same number 3, so Tool #15 says: rewrite them in one common format — powers of 3 — and the hidden structure comes out. Then Tool #4 names the missing quantity, the common ratio r, so the definition of a geometric progression can be applied instead of guessed at: compute r from the first two terms, confirm the same r carries the second term to the third, and only then step forward to the fourth. Tool #5 supplies a second, independent reading — the exponents form an arithmetic sequence — used as a cross-check, not as the proof, because three numbers alone never force a pattern. Tool #3 closes the loop against the answer list, which is deliberately built to reward a pattern read off the wrong sequence.
Put every term on base 3
Every term becomes a power of one number.
Once every term is a power of the same base, multiplying and dividing terms turns into adding and subtracting exponents.
Once every term is a power of the same base, multiplying and dividing turn into adding and subtracting exponents.
▸ Why?
An exponent counts how many times the base is used, so combining powers combines those counts.
▸ Why?
A geometric run multiplies by the same number each step, which in exponents is the same step added each time.
Give the exponents one denominator
One denominator makes the exponents comparable.
A common denominator turns three unrelated-looking fractions into the plain countable list 3/6, 2/6, 1/6.
7.NS.A.1Organize Information In More WaysFind the ratio and verify it
The exponents step down by a constant.
Two consecutive ratios that agree is exactly what "geometric" means — checking them is how the fourth term becomes forced instead of guessed.
11.N-RN.A.1Introduce A VariableTake one more step
One more step lands the exponent at zero.
The exponent walks down by 1/6 every step, and the fourth step lands exactly on 0 — the power that means "no factors of 3 left at all".
9.F-LE.A.2Look For A PatternCheck against the choices
So the next term is 1, choice (A).
Every radical on the list is a power of 3 with a positive exponent, so none of them can be the term whose exponent has been driven down to 0.
11.N-RN.A.2Eliminate PossibilitiesWrite every root as a power of the same base, then "geometric" simply means the exponents step down by the same amount each time — here the fourth step lands on 3⁰ = 1.
- Put every term on base 3
- Give the exponents one denominator
- Find the ratio and verify it
- Take one more step
- Check against the choices