A designer creates a sequence of mosaic patterns using modular tile units. The table below shows the number of tile...
GMAT Advanced Math : (Adv_Math) Questions
A designer creates a sequence of mosaic patterns using modular tile units. The table below shows the number of tile units used in the first three patterns:
| Pattern | Tile Units |
|---|---|
| 1 | \(\mathrm{r}\) |
| 2 | \(\mathrm{r}^5\) |
| 3 | \(\mathrm{r}^9\) |
For this exponential mosaic sequence, \(\mathrm{r}\) is a constant greater than 1. If Pattern \(\mathrm{k}\) uses \(\mathrm{r}^{29}\) tile units, what is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Pattern 1 uses \(\mathrm{r^1}\) tile units
- Pattern 2 uses \(\mathrm{r^5}\) tile units
- Pattern 3 uses \(\mathrm{r^9}\) tile units
- Pattern k uses \(\mathrm{r^{29}}\) tile units
- Need to find: the value of k
2. INFER the pattern in exponents
- Look at the exponents: 1, 5, 9
- Find the differences: \(\mathrm{5-1 = 4}\), \(\mathrm{9-5 = 4}\)
- Since the difference is constant (4), this suggests an arithmetic pattern
- For pattern n, the exponent appears to follow: \(\mathrm{4n - 3}\)
3. INFER verification of the pattern formula
- Check if \(\mathrm{4n - 3}\) works:
- Pattern 1: \(\mathrm{4(1) - 3 = 1}\) ✓
- Pattern 2: \(\mathrm{4(2) - 3 = 5}\) ✓
- Pattern 3: \(\mathrm{4(3) - 3 = 9}\) ✓
- The pattern holds!
4. TRANSLATE the target condition
- Pattern k uses \(\mathrm{r^{29}}\) tile units
- This means the exponent for pattern k equals 29
- So: \(\mathrm{4k - 3 = 29}\)
5. SIMPLIFY to solve for k
- \(\mathrm{4k - 3 = 29}\)
- \(\mathrm{4k = 29 + 3}\)
- \(\mathrm{4k = 32}\)
- \(\mathrm{k = 8}\)
Answer: 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students see the exponents 1, 5, 9 but don't recognize the arithmetic pattern or incorrectly identify the pattern rule.
They might think the pattern is "add 4 each time" and incorrectly conclude that pattern 4 would have exponent 13, pattern 5 would have exponent 17, etc. This leads to setting up the wrong equation and getting an incorrect value for k. This causes them to get stuck and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the pattern formula \(\mathrm{4n - 3}\) and set up \(\mathrm{4k - 3 = 29}\), but make arithmetic errors when solving.
Common mistakes include: \(\mathrm{4k = 29 - 3 = 26}\), giving \(\mathrm{k = 6.5}\), or other calculation errors. Since the answer must be a whole number (pattern numbers are integers), this leads to confusion and guessing.
The Bottom Line:
This problem tests pattern recognition skills more than complex mathematical concepts. Students must see past individual data points to identify the underlying arithmetic structure, then apply it systematically.