Question

Complete the given series by finding the missing term-
0, 14, 78, 252, 620, —

• A. 2387
• B. . 2394
• C. 2400
• D. None of the above

Hint:

When analyzing number series: \begin{itemize} \item \textbf{Calculate Differences:} Find the differences between consecutive terms. If the first differences are not constant, find the differences of the differences (second differences), and so on. \item \textbf{Identify Constant Difference Order:} If a certain order of differences becomes constant, the series is a polynomial. For a constant $k$-th difference, the polynomial is of degree $k$. The leading coefficient $A = \frac{\text{Constant } k\text{-th Difference}}{k!}$. \item \textbf{Hypothesize and Test:} Based on the growth rate, try to hypothesize a pattern involving squares, cubes, or powers of $n$. Then, test this hypothesis with the given terms. Look for simple adjustments (e.g., $n^2+c$, $n^3-c$, etc.). \end{itemize}

Answer 1

The Correct Option is D

Step 1: Analyze the differences between consecutive terms to find a pattern.
Given series: $0, 14, 78, 252, 620, \text{X}$
First Differences:
$14 - 0 = 14$
$78 - 14 = 64$
$252 - 78 = 174$
$620 - 252 = 368$
The first differences are: $14, 64, 174, 368$
Second Differences:
$64 - 14 = 50$
$174 - 64 = 110$
$368 - 174 = 194$
The second differences are: $50, 110, 194$
Third Differences:
$110 - 50 = 60$
$194 - 110 = 84$
The third differences are: $60, 84$
Fourth Differences:
$84 - 60 = 24$
The fourth difference is a constant: $24$.
Step 2: Determine the general term ($T_n$) of the series.
Since the fourth difference is constant, the series follows a polynomial pattern of degree 4. The general term can be expressed as $T_n = An^4 + Bn^3 + Cn^2 + Dn + E$.
The coefficient $A$ is found by $A = \frac{\text{Constant Fourth Difference}}{4!} = \frac{24}{24} = 1$.
So, the formula for the $n$-th term starts with $n^4$.
Let's test the hypothesis $T_n = n^4 - n$:
For $n=1$: $T_1 = 1^4 - 1 = 1 - 1 = 0$ (Matches the first term)
For $n=2$: $T_2 = 2^4 - 2 = 16 - 2 = 14$ (Matches the second term)
For $n=3$: $T_3 = 3^4 - 3 = 81 - 3 = 78$ (Matches the third term)
For $n=4$: $T_4 = 4^4 - 4 = 256 - 4 = 252$ (Matches the fourth term)
For $n=5$: $T_5 = 5^4 - 5 = 625 - 5 = 620$ (Matches the fifth term)
The pattern $T_n = n^4 - n$ is consistent with all given terms.
Step 3: Calculate the missing term.
The missing term is the 6th term in the series (for $n=6$).
$T_6 = 6^4 - 6$
$T_6 = 1296 - 6$
$T_6 = 1290$
Step 4: Compare the result with the given options.
The calculated missing term is 1290.
A. 2387
B. 2394
C. 2400
D. None of the above
Since 1290 is not listed in options A, B, or C, the correct choice is D. The final answer is $\boxed{\text{1290}}$.