Find the missing number in the series 15, 51, 216, 1100, ?, 46452.

Introduction / Context:

This number series grows very rapidly and is constructed from a more sophisticated rule that combines multiplication with an additive term that is itself structured. The problem checks higher level numerical reasoning and the ability to detect patterns that incorporate both the position of the term and its value. Such questions often appear in more challenging sections of reasoning tests.

Given Data / Assumptions:

• The given series is 15, 51, 216, 1100, ?, 46452.
• One number between 1100 and 46452 is missing.
• The progression appears to involve multiplying by increasing integers.
• There is also an added term that depends on the same integer used for multiplication.

Concept / Approach:

A natural strategy is to test whether a(n+1) can be expressed as a(n) multiplied by a small integer plus some structured additive term. Examining the first few transitions can reveal how both the multiplier and the addition depend on the step number. Here, we find that the multiplier grows in a simple linear way, and the additive term is the product of two consecutive integers. Once we find a general rule that matches all known transitions, we can apply it to calculate the missing term and confirm it by checking the final transition to 46452.

Step-by-Step Solution:

Step 1: From 15 to 51: 15 * 3 = 45, and 45 + 6 = 51; note that 6 = 3 * 2. Step 2: From 51 to 216: 51 * 4 = 204, and 204 + 12 = 216; note that 12 = 4 * 3. Step 3: From 216 to 1100: 216 * 5 = 1080, and 1080 + 20 = 1100; note that 20 = 5 * 4. Step 4: The pattern is: a(n+1) = a(n) * k + k * (k - 1), where k takes values 3, 4, 5 and increases by 1 each time. Step 5: For the next step, k = 6, so the missing term is a(5) = 1100 * 6 + 6 * 5 = 6600 + 30 = 6630. Then with k = 7, compute 6630 * 7 + 7 * 6 = 46410 + 42 = 46452, matching the given last term.

Verification / Alternative check:

We can describe the rule generically as a(n+1) = a(n) * (n + 2) + (n + 2) * (n + 1), starting from n = 1. Plugging in n = 1, we get 15 * 3 + 3 * 2 = 45 + 6 = 51. For n = 2, 51 * 4 + 4 * 3 = 204 + 12 = 216. For n = 3, 216 * 5 + 5 * 4 = 1080 + 20 = 1100. For n = 4, 1100 * 6 + 6 * 5 = 6600 + 30 = 6630. For n = 5, 6630 * 7 + 7 * 6 = 46410 + 42 = 46452. The rule holds at every step, confirming both the pattern and the missing value 6630.

Why Other Options Are Wrong:

Option 5660 would not satisfy the rule, because 5660 * 7 + 7 * 6 does not equal 46452. Option 6560 similarly fails when used in the formula. Option 6750 produces 6750 * 7 + 42 = 47292, which does not match the given final term. Option 6000 also does not lead to 46452 using the same pattern. Only 6630 simultaneously preserves the correct relationship with 1100 and yields 46452 when we apply the next step of the rule.

Common Pitfalls:

Many learners either ignore the additive term and search for pure multiplication, or they try to derive a pattern only from the differences. Both approaches fail because neither the ratios nor the differences alone follow a simple rule. Another mistake is to give up after matching only the first couple of transitions and not verifying the pattern against later terms. In high level series, confirming that the final given term also fits the same rule is crucial.

Final Answer:

The missing number that makes every transition conform to the rule a(n+1) = a(n) * k + k * (k - 1) is 6630.