Practise the method.
Example 1
Pattern A follows A(n)=n2+bn+c. Its first two stages have 48 and 58 tiles. Pattern B starts with 24 tiles and adds 19 each stage. At which two positive stages do the patterns have equal tile counts?
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Answer: x=5x=7
- For A, subtract the first two values to get b=7
- c=40.
- B(n)=19n+5.
- Equating gives n2−12n+35=0.
- Hence the equal-count stages are 5 and 7.
Example 2
A tile pattern follows T(n)=n2+bn+c, with constant b and c. Stage 5 uses 64 tiles and stage 7 uses 102. Which positive stage uses 232 tiles?
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Answer: 12
- Subtract the equations and remove the known difference of squares: 2b=(102−64)−(72−52)=14
- b=7.
- Substituting gives c=4.
- Now solve n2+7n+4=232.
- The positive stage is 12;
- the other root is−19.
Example 3
Pattern A follows A(n)=n2+bn+c. Its first two stages have 25 and 35 tiles. Pattern B starts with 19 tiles and adds 16 each stage. At which two positive stages do the patterns have equal tile counts?
Show worked answer
Answer: x=2x=7
- For A, subtract the first two values to get b=7
- c=17.
- B(n)=16n+3.
- Equating gives n2−9n+14=0.
- Hence the equal-count stages are 2 and 7.