Question

Find the missing number in the following series :
2345, 2165, 2065, 2017, ?, 1995, 1995

• A. 1990
• B. 2005
• C. 2102
• D. 1999

Hint:

If the differences of a series do not form a simple arithmetic progression, try writing each difference as a function of its index:
Common patterns include $n^2 + n$, $n^2 - n$, $n^3 + n$, or $n^3 - n^2$.
Recognizing numbers like $180$, $100$, and $48$ as near perfect cubes ($216, 125, 64$) can quickly reveal the $n^3 - n^2$ pattern.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This is a number series problem where we need to find the logical pattern governing the transition from one term to the next.
By calculating the differences between successive terms, we can identify if there is a arithmetic or geometric progression, or if the differences follow a specific mathematical function (such as squares or cubes).

Step 2: Detailed Explanation:

• Calculate Differences Between Known Consecutive Terms:
Let us list the terms of the series and find the differences:
- Difference 1: $2345 - 2165 = 180$
- Difference 2: $2165 - 2065 = 100$
- Difference 3: $2065 - 2017 = 48$
- Let the missing term be $x$.
- Difference 4: $2017 - x$
- Difference 5: $x - 1995$
- Difference 6: $1995 - 1995 = 0$
The sequence of differences is: $180, 100, 48, \dots, 0$.

• Analyze the Pattern of Differences:
Let us analyze the numbers $180, 100, 48$:
- For $n = 6$:
\[ n^3 - n^2 = 6^3 - 6^2 = 216 - 36 = 180 \]
- For $n = 5$:
\[ n^3 - n^2 = 5^3 - 5^2 = 125 - 25 = 100 \]
- For $n = 4$:
\[ n^3 - n^2 = 4^3 - 4^2 = 64 - 16 = 48 \]
The pattern for the difference subtracted to get the next term is $n^3 - n^2$ for decreasing integers $n = 6, 5, 4, 3, 2, 1$.

• Apply the Pattern to Find the Missing Number:
For $n = 3$, the difference should be:
\[ 3^3 - 3^2 = 27 - 9 = 18 \] Therefore, the missing term $x$ is:
\[ x = 2017 - 18 = 1999 \]

• Verify with the Remaining Terms:
Let us check the next differences with $x = 1999$:
- For $n = 2$:
The difference should be:
\[ 2^3 - 2^2 = 8 - 4 = 4 \] Check: $1999 - 4 = 1995$ (Matches the next term!).
- For $n = 1$:
The difference should be:
\[ 1^3 - 1^2 = 1 - 1 = 0 \] Check: $1995 - 0 = 1995$ (Matches the final term!).
The pattern is correct and verified.

Step 3: Final Answer:
The missing number in the series is 1999.
Therefore, the correct option is (D).