Question

Questions 43 and 44 are based on the following instructions: each question gives a statement followed by three conclusions, I, II and III. Pick the option that tells you which of the three conclusions can be derived from the statement alone.

Statement: \(A_0, A_1, A_2, \ldots\) is a sequence of numbers with \(A_0 = 1\), \(A_1 = 3\), and \(A_t = (t+1)A_{t-1} - tA_{t-2}\) for \(t = 2, 3, 4, \ldots\)

Conclusion I: \(A_8 = 77\)
Conclusion II: \(A_{10} = 121\)
Conclusion III: \(A_{12} = 145\)

• A. Using the given statement, only conclusion I can be derived.
• B. Using the given statement, only conclusion II can be derived.
• C. Using the given statement, only conclusion III can be derived.
• D. Using the given statement, none of the three conclusions I, II and III can be derived.

Hint:

Look at the difference \(A_t - A_{t-1}\) between consecutive terms rather than recomputing every term from the recurrence; that difference follows a much simpler pattern than the original recurrence does.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept.
Instead of hunting for a general formula, just work out the sequence one term at a time straight from the recurrence, and check whether the given conclusions match.

Step 2: Compute the first few terms directly.
We know $A_0 = 1$ and $A_1 = 3$. For $t = 2$: $A_2 = (2+1)A_1 - 2A_0 = 3(33) - 2(11) = 9 - 2 = 7$.
For $t = 3$: $A_3 = 4A_2 - 3A_1 = 4(77) - 3(33) = 28 - 9 = 19$.
For $t = 4$: $A_4 = 5A_3 - 4A_2 = 5(19) - 4(77) = 95 - 28 = 67$.
For $t = 5$: $A_5 = 6A_4 - 5A_3 = 6(67) - 5(19) = 402 - 95 = 307$.

Step 3: Keep applying the recurrence up to $t = 8$.
For $t = 6$: $A_6 = 7A_5 - 6A_4 = 7(307) - 6(67) = 2149 - 402 = 1747$.
For $t = 7$: $A_7 = 8A_6 - 7A_5 = 8(1747) - 7(307) = 13976 - 2149 = 11827$.
For $t = 8$: $A_8 = 9A_7 - 8A_6 = 9(11827) - 8(1747) = 106443 - 13976 = 92467$.
This already tells us $A_8 = 92467$, nowhere near the value $77$ claimed in Conclusion I.

Step 4: Push the computation on to $t = 10$ and $t = 12$.
For $t = 9$: $A_9 = 10A_8 - 9A_7 = 10(92467) - 9(11827) = 924670 - 106443 = 818227$.
For $t = 10$: $A_{10} = 11A_9 - 10A_8 = 11(818227) - 10(92467) = 9000497 - 924670 = 8075827$.
For $t = 11$: $A_{11} = 12A_{10} - 11A_9 = 12(8075827) - 11(818227) = 96909924 - 9000497 = 87909427$.
For $t = 12$: $A_{12} = 13A_{11} - 12A_{10} = 13(87909427) - 12(8075827) = 1142822551 - 96909924 = 1045912627$.

Step 5: Match against the conclusions and pick the option.
$A_8 = 92467 \neq 77$, so Conclusion I is false.
$A_{10} = 8075827 \neq 121$, so Conclusion II is false.
$A_{12} = 1045912627 \neq 145$, so Conclusion III is false.
Since term by term computation rules out all three conclusions, none of them can be derived from the statement.
This matches option (EE), the same answer reached through the shortcut factorial formula, which confirms the result is not a one-off calculation slip.