AMC 10 · 2009 · #12
Grade 8 algebraPick an answer.
Tool #4 (Introduce a Variable) names the two things that describe the whole sequence: the first term a and the ratio r. Then the work splits into two independent subproblems (tool #7): find r, then find a. The ratio falls out because the fifth and eighth terms are three steps apart, so dividing them cancels a entirely, and the right-hand side is 8!/7! = 8 straight from the definition of factorial. Tool #3 (Eliminate Possibilities) does the part the quick solution skips — showing r = 2 is the only real cube root of 8, which is exactly where the words "of real numbers" earn their place. Tool #6 (Guess and Check) finishes by building the sequence and confirming it really has the two required terms.
Name the first term and the ratio
Naming the first term and ratio writes both given terms.
Where you start and what you multiply by are the only two facts a geometric sequence has, so name them and let the two given terms pin them down.
6.EE.A.2Introduce A VariableDivide to cancel the first term
Dividing cancels the first term and the factorials.
Dividing two terms of a geometric sequence erases the starting value and leaves only the ratio raised to the number of steps between them.
Dividing two terms of a geometric sequence erases the starting value and leaves only the ratio to a power.
▸ Why?
Every term is the first one multiplied by the ratio some number of times, so the first term is a shared factor.
▸ Why?
An exponent counts how many times the ratio has been used, so the quotient's exponent is the gap between the positions.
Why r has to be exactly 2
Factoring shows the ratio is exactly two.
Unlike squaring, cubing never folds the number line back on itself, so a real cube root is unique.
8.EE.A.2Eliminate PossibilitiesBack out the first term
Backing out gives the first term as 315.
Once the ratio is known, the trip from the first term to the fifth is four doublings, so one division by 16 walks it back.
6.EE.B.7Introduce A VariableCheck the sequence actually exists
Writing the sequence out confirms it, choice (E).
Solving equations shows what the answer would have to be; building the sequence shows that it is really there.
4.OA.C.5Guess And CheckDividing a later term of a geometric sequence by an earlier one wipes out the starting value, so the gap between two known terms tells you exactly what you are multiplying by.
- Name the first term and the ratio
- Divide to cancel the first term
- Why r has to be exactly 2
- Back out the first term
- Check the sequence actually exists