Directions: Find the wrong number in the series. 3, 4, 9, 22.5, 67.5, 202.5, 810

### Concept & Logic The presence of decimals ($22.5, 67.5$) strongly suggests multiplication by fractions or decimals, often starting with $0.5, 1.5, 2.5$ etc., or a consistent fractional multiplier. ### Step-by-Step Solution 1. Let's look at the later terms where the relationship might be clearer. 2. $202.5 / 67.5 = 3$ 3. $67.5 / 22.5 = 3$ 4. It seems there is a constant multiplier of $3$ in the latter part of the series. Let's check the rest. 5. Wait, $810 / 202.5 = 4$. So the multiplier is not constant. 6. Let's re-examine the multipliers: $\times 3$, $\times 3$, $\times 4$. This doesn't seem like a smooth pattern. 7. Let's try another common pattern: increasing decimal multipliers. Maybe $\times 1.5, \times 2, \times 2.5, \times 3$? Let's test this from the beginning. $3 \times 1 = 3$ (Given is $4$. Let's assume $4$ is wrong and should be $3$, or the pattern starts differently). Let's try multipliers: $1.5, 2.5, 3.5$ etc. 8. Let's look at the transitions again. $22.5 \times 3 = 67.5$ $67.5 \times 3 = 202.5$ $202.5 \times 4 = 810$ This is strange. Let's try $\times 1, \times 1.5, \times 2, \times 2.5$... If the series is $3, 3, 4.5, 9, 22.5...$ that doesn't match. 9. Let's try $\times 1.5, \times 2, \times 2.5$ starting from a different point. $9 \times 2.5 = 22.5$ $22.5 \times 3 = 67.5$ $67.5 \times 3 = 202.5$ (Still $\times 3$. Breaks a simple increasing pattern). 10. Let's try to find a pattern in $9, 22.5, 67.5, 202.5, 810$. $9 \times 2.5 = 22.5$ $22.5 \times 3 = 67.5$ $67.5 \times 3 = 202.5$ $202.5 \times 4 = 810$ Maybe the multipliers are $1.5, 2, 2.5, 3, 3.5$? Let's trace backwards from $810$. $810 / 4 = 202.5$ $202.5 / 3 = 67.5$ $67.5 / 3 = 22.5$ (Pattern of multipliers: $...3, 3, 4$. Not consistent.) 11. Let's try another approach. Look at the ratio of consecutive terms. $x_n / x_{n-1}$. $22.5 / 9 = 2.5$ $67.5 / 22.5 = 3$ $202.5 / 67.5 = 3$ $810 / 202.5 = 4$ There might be an error in my assumption or the question itself. Let's reconsider the sequence: $3, 4, 9, 22.5, 67.5, 202.5, 810$. Let's try $3 \times 1.5 = 4.5$. $4.5 \times 2 = 9$. $9 \times 2.5 = 22.5$. $22.5 \times 3 = 67.5$. $67.5 \times 3 = 202.5$ --- WAIT! $67.5 \times 3.5 = 236.25$. Let me re-read the numbers. $3, 4, 9, 22.5, 67.5, 202.5, 810$. Ah, let's check $202.5 \times 4$. $200 \times 4 = 800$, $2.5 \times 4 = 10$. So $810$. The multipliers are: $?, ?, 2.5, 3, 3, 4$. This is very irregular. 12. Let's try: $\times 1.5, \times 2, \times 2.5, \times 3, \times 4$? Let's reconsider the initial terms. $3 \times 1 = 3 + 1 = 4$ $4 \times 2 = 8 + 1 = 9$ $9 \times 2.5 = 22.5$ (addition pattern breaks) 13. Let's try: $3 \times 1.5 = 4.5$ $4.5 \times 2 = 9$ $9 \times 2.5 = 22.5$ $22.5 \times 3 = 67.5$ $67.5 \times 3.5 = 236.25$ (Given is $202.5$). Wait, let's look at the options. Option (e) is $202.5$. But let's check the next term if $202.5$ was $236.25$. $236.25 \times 4 = 945$. Given is $810$. What if the wrong term is $4$? If series is $3, 4.5, 9, 22.5, 67.5...$ Multipliers: $1.5, 2, 2.5, 3$. Then next is $67.5 \times 3 = 202.5$. Why $\times 3$ again? Let's re-calculate $67.5 \times 3$. Yes, it's $202.5$. Maybe the pattern of multipliers is $1.5, 2, 2.5, 3, 4$? That doesn't make sense. Let's reconsider the multipliers: $22.5 / 9 = 2.5$ $67.5 / 22.5 = 3$ $202.5 / 67.5 = 3$ $810 / 202.5 = 4$ This is highly irregular. Let me re-read the image carefully. $3, 4, 9, 22.5, 67.5, 202.5, 810$. Is it possible the multiplier is $\times 1.5, \times 2, \times 2.5, \times 3, \times 3.5, \times 4$? Let's try to fix it. If the multipliers should be $1.5, 2, 2.5, 3, 4$ maybe? No. Let's look at: $3, 4.5, 9, 22.5, 67.5, 236.25, 945$. That would mean two wrong numbers. Let's re-examine: $67.5 \times 3 = 202.5$. $202.5 \times 4 = 810$. Multipliers: $?, ?, 2.5, 3, 3, 4$. What if the series is $3 \times 1 = 3$? No. Let's try $2, 4, 9...$ No. Let's look at the options again. $4, 9, 22.5, 67.5, 202.5$. Suppose the pattern is $\times 1.5, \times 2, \times 2.5, \times 3$. $x \times 1.5 = y$ $y \times 2 = 9 \Rightarrow y = 4.5$ $x \times 1.5 = 4.5 \Rightarrow x = 3$. So the beginning of the series should be $3, 4.5, 9, 22.5, 67.5$. The given series has $4$ instead of $4.5$. This makes $4$ the wrong number. What about the rest of the series? $67.5, 202.5, 810$. $67.5 \times 3 = 202.5$. $202.5 \times 4 = 810$. The multipliers are $1.5, 2, 2.5, 3, 3, 4$? This seems flawed. Let me re-read the options. Maybe I misread a number. $3, 4, 9, 22.5, 67.5, 202.5, 810$. Let's reconsider the multiplier pattern. Maybe it's $\times 1, \times 1.5, \times 2, \times 2.5, \times 3, \times 3.5...$ $3 \times 1 = 3$ (If $4$ is wrong, maybe the second term should be $3$?) $3 \times 1.5 = 4.5 \neq 9$ Let's try another pattern. Maybe difference? $1, 5, 13.5, 45...$ no. Let's stick with the most promising one: $3 \times 1.5 = 4.5$. $4.5 \times 2 = 9$. $9 \times 2.5 = 22.5$. $22.5 \times 3 = 67.5$. If $4$ is the wrong number, the first few terms work beautifully. The latter terms ($202.5, 810$) might just have irregular multipliers in the question setter's mind, or there's a typo in the question itself for the later terms. Given the options, $4$ is the earliest and most likely intended anomaly if we assume a growing multiplier pattern. Let's assume the intended multipliers were $1.5, 2, 2.5, 3, 3.5, 4$. Then $67.5 \times 3.5 = 236.25 \neq 202.5$. Let's try another pattern. $3 \times 0.5 + 2.5 = 4$. $4 \times 1 + 5 = 9$. $9 \times 1.5 + 9 = 22.5$. $22.5 \times 2 + 22.5 = 67.5$. $67.5 \times 2.5 + ... = 202.5$. Let's calculate $67.5 \times 2.5 = 168.75$. $202.5 - 168.75 = 33.75$. Additions: $2.5, 5, 9, 22.5, 33.75$. No obvious pattern. Let's go back to $3 \times 1.5 = 4.5$. The error is right at the beginning. In many such series problems, finding the first clear break in a logical pattern points to the answer. The pattern $\times 1.5, \times 2, \times 2.5$ for $3, (4.5), 9, 22.5$ is very strong. ### Exam Strategy & Shortcut Spotting decimals like $.5$ in the middle of a series is a strong indicator of fractional multipliers ($\times 1.5, \times 2.5$, etc.). Test this hypothesis starting from a pair that works, like $9$ and $22.5$ ($9 \times 2.5 = 22.5$). Work backwards to find the early error. ### Common Pitfall Trying to establish an additive pattern when decimals appear in a sequence that is growing significantly. ### Final Answer Therefore, the correct answer is **4**.