What will come in place of the question mark in the following number series: 5, 14, 45, 130, ?, 1182?

Introduction / Context:

This number series problem features rapidly increasing values where the pattern is not simply multiplication by a fixed factor. Instead, a combination of multiplication and addition or subtraction is involved, with the additive part following its own structure. Such series are designed to measure a candidate ability to handle multi layer patterns, where both the multiplicative factor and the additive or subtractive term change in a controlled way across the sequence.

Given Data / Assumptions:

• The series is 5, 14, 45, 130, ?, 1182.
• One term between 130 and 1182 is missing.
• The growth between terms is steep, suggesting repeated multiplication by a similar factor, with extra adjustments.
• The adjustments are likely to be systematically increasing odd numbers with alternating signs.

Concept / Approach:

A common pattern in advanced series involves multiplying by a constant factor, such as 3, and then adding or subtracting successive odd numbers. Here, we suspect a rule like a(n+1) = 3 * a(n) + or minus some odd number following a simple pattern. We must check each step, determine the sequence of these odd numbers, and see how they are combined with multiplication by 3. Once that is identified, we can compute the missing term and verify that the subsequent given term also fits the same rule.

Step-by-Step Solution:

Step 1: From 5 to 14: 5 * 3 = 15; to get 14 we subtract 1, so 5 * 3 - 1 = 14. Step 2: From 14 to 45: 14 * 3 = 42; to get 45 we add 3, so 14 * 3 + 3 = 45. Step 3: From 45 to 130: 45 * 3 = 135; to get 130 we subtract 5, so 45 * 3 - 5 = 130. Step 4: The adjustment terms are -1, +3, -5, which are odd numbers with alternating signs and increasing magnitude. The next adjustment should be +7. Step 5: Therefore, the missing term after 130 is 130 * 3 + 7 = 390 + 7 = 397. For verification, 397 * 3 = 1191, and the next adjustment must be -9, giving 1191 - 9 = 1182, which matches the given last term.

Verification / Alternative check:

Summarize the rule as a(n+1) = 3 * a(n) + k(n), with k(n) being -1, +3, -5, +7, -9 and so on, that is alternating signs with odd numbers in ascending order. Check each step: 5 * 3 - 1 = 14, 14 * 3 + 3 = 45, 45 * 3 - 5 = 130, 130 * 3 + 7 = 397, 397 * 3 - 9 = 1182. All transitions match both the multiplicative factor 3 and the odd number adjustment sequence, so the pattern is internally consistent and confirms 397 as the correct missing term.

Why Other Options Are Wrong:

Option 222 does not work because 130 * 3 plus or minus any single odd number cannot produce 222, and 222 * 3 plus or minus an odd number will not yield 1182. Option 404 also fails both from 130 and towards 1182. Option 415 does not fit, since 130 * 3 is 390, and adding odd numbers 1, 3, 5, 7, 9 cannot give 415 without breaking the alternating sign rule. Option 350 similarly cannot be reconciled with the established sequence of adjustments. Only 397 satisfies both transitions, from 130 and to 1182, under the same rule.

Common Pitfalls:

Learners often attempt to look at simple differences between terms in such non linear series, but the differences 9, 31, 85, and so on appear quite irregular. Another error is to guess that the series is purely geometric, which does not hold here because the ratio between terms is not constant. Some students also miss the alternating sign pattern in the adjustments and treat them as random. Recognizing the repeated factor 3 and then focusing on the sequence of odd numbers is the key to solving this efficiently.

Final Answer:

The number that correctly completes the pattern of multiplying by 3 and adding or subtracting successive odd numbers is 397.