Calculate the value of (A) and (B) and answer the questions given below: (A), (B), 102, 104, 107, 113, 128, 173 If M is 150% more than (A), then find which of the following number is nearest square root of the difference between M and (B).

### Concept & Pattern Recognition To determine the unknown variables in a sequence, extract the differences between the known terms and identify a secondary pattern (such as a multiplicative factor) that defines these differences. ### Step-by-Step Solution * **Identify Differences:** For the known sequence $102, 104, 107, 113, 128, 173$, the differences are: * $104 - 102 = 2$ * $107 - 104 = 3$ * $113 - 107 = 6$ * $128 - 113 = 15$ * $173 - 128 = 45$ * **Analyze Double Pattern:** Look at the ratio of these consecutive differences: * $3 / 2 = 1.5$ * $6 / 3 = 2$ * $15 / 6 = 2.5$ * $45 / 15 = 3$ * The multipliers are increasing by $0.5$ ($1.5, 2.0, 2.5, 3.0$). * **Find (A) and (B):** Working backward, the multiplier before $1.5$ must be $1.0$, and before that $0.5$. * Let the difference between $102$ and $(B)$ be $d_2$. Then $d_2 \times 1 = 2 \Rightarrow d_2 = 2$. * So, $102 - (B) = 2 \Rightarrow (B) = 100$. * Let the difference between $(B)$ and $(A)$ be $d_1$. Then $d_1 \times 0.5 = 2 \Rightarrow d_1 = 4$. * So, $(B) - (A) = 4 \Rightarrow 100 - (A) = 4 \Rightarrow (A) = 96$. * **Calculate M:** $M$ is $150\%$ more than $(A)$, meaning $M = (A) + 1.5(A) = 2.5 \times 96 = 240$. * **Final Calculation:** The difference between $M$ and $(B)$ is $240 - 100 = 140$. * The square root of $140$ is approximately $11.83$. The nearest integer is $12$. ### Exam Strategy & Shortcut When you see numbers growing slowly and then rapidly (like differences jumping from 2 to 45), suspect a multiplicative pattern within the differences. Recognizing this quickly saves time trying additive or exponential patterns. ### Common Pitfall A frequent error is misinterpreting "150% more than (A)" as just "$1.5 \times A$". "More than" means you must add it to the original value, making it $A + 1.5A = 2.5A$. ### Final Answer Therefore, the correct answer is **12**.