AMC 10 · 2004 · #10
Grade 6 arithmeticPick an answer.
The natural instinct is to name the first term, but then the sum drags in 1+2+…+48. Tool #4 with a better choice of variable — name the middle term instead — makes the list symmetric about the unknown, and Tool #15 (writing the list centred on its median rather than starting from its front) makes the outer terms cancel in pairs. The sum then collapses to 49 copies of the middle term, and one division finishes the problem. Tool #3 is the last step: the choices are all powers of 7, so the numerical answer has to be matched back to a power.
Locate the median in the list
With an odd count the median is a real member — here the 25th term.
An odd-length ordered list has one true middle, so the median is a term you can point at.
6.SP.A.3Organize Information In More WaysName the middle term
Naming it and writing the rest as offsets centres the whole list.
Naming the unknown you actually want, instead of the first term, is what makes the list line up symmetrically.
6.EE.B.6Introduce A VariableAdd by pairing outward from the middle
Pairing outward cancels every offset, so the sum is just 49 times the median.
In a symmetric list the amounts above the middle exactly cancel the amounts below it, leaving only the middle counted once per term.
Adding outward from the middle, the amounts above cancel the amounts below and only the middle survives per term.
▸ Why?
Terms placed symmetrically about the middle always pair to the same total, whatever the spacing.
▸ Why?
That common total is twice the middle, so the middle is the level the whole list would sit at after fair sharing.
Solve for the middle term
Dividing the given total gives 343.
If 49 equal shares make the total, one share is the total divided by 49.
6.NS.B.2Introduce A VariableMatch 343 to a power of 7
Written as a power of seven that is 7³, choice (C).
When the choices are all powers of one number, rewrite your answer as a power of that number too.
6.EE.A.1Eliminate PossibilitiesIn a list of consecutive numbers the middle one is also the average, so the sum is just the count times the median — divide 16807 by 49 and you get 343.
- Locate the median in the list
- Name the middle term
- Add by pairing outward from the middle
- Solve for the middle term
- Match 343 to a power of 7