Given below are two number series I and II where the missing numbers in series I and II are 'P' and 'Q', respectively. Find the value of 'P' and 'Q' and find which among the given... Given below are two number series I and II where the missing numbers in series I and II are 'P' and 'Q', respectively. Find the value of 'P' and 'Q' and find which among the given options gives the correct value of (P - Q). I: 5000, 4000, 3000, 2100, P, 819 II: 4, 6, 15, 56, Q, 1644 [1] 1120 [2] 1090 [3] 980 [4] 1240 [5] 1050 Read more

Steps to Solve

1. Identify the pattern in Series I

   The first series is: $5000, 4000, 3000, 2100, P, 819$.
   To find the differences:

   • $5000 - 4000 = 1000$
   • $4000 - 3000 = 1000$
   • $3000 - 2100 = 900$

   The differences are decreasing by 100 each time until now.
   Following this pattern, the next difference would be $2100 - (900 - 100) = 2100 - 800 = 1300$.
   Thus,
   $$ P = 2100 - 800 = 1300 $$

2. Identify the pattern in Series II

   The second series is: $4, 6, 15, 56, Q, 1644$.
   To find the ratios of consecutive numbers:

   • $6 = 4 \times 1.5$
   • $15 = 6 \times 2.5$
   • $56 = 15 \times \frac{56}{15} \approx 3.73$ (this suggests a growing difference)

   Instead of a fixed ratio, we look for a pattern in differences:

   • $6 - 4 = 2$
   • $15 - 6 = 9$
   • $56 - 15 = 41$

   The differences seem to escalate quickly. Finding the next term:

   • Assuming a pattern in the differences:
   • $9 - 2 = 7$
   • $41 - 9 = 32$
   • The next difference might be around 41 + (another growth factor). Given the large jump to $Q$ and $1644$, assume it's quite large and verify against the next known value, $1644$.

3. Estimate Q

   If we try $Q$ around, let's consider it conservatively around $800$ to fit somewhere logically, and validate back:

   • Check: $56 + d$ leading closer to $1644$, fitting possibly with exponential jumps or multiplicative terms.

   Further analysis could yield the exact, but using estimates or small guess-and-check can settle.

4. Calculating the Difference (P - Q)

   Once $P = 1300$ and assume final calculated $Q = 819$ (or appropriate from estimations), subtract them: $$ P - Q = 1300 - Q $$
   With $Q$, deduce further values depending on function fitting.