Find the missing number in the series: 1, 8, 36, ?, 1100, 7701.

Introduction / Context:
This number series is based on a pattern where each term is obtained by multiplying the previous term by an increasing integer and then adding that same integer. This is similar in spirit to some earlier sequences but uses larger multipliers and addends, causing rapid growth in the numbers.

Given Data / Assumptions:

• Series: 1, 8, 36, ?, 1100, 7701.
• One term (the fourth term) is missing.
• The large jumps suggest multiplicative steps combined with additions.

Concept / Approach:
Because the numbers grow quickly, we attempt a pattern where each term equals the previous term multiplied by an integer n, plus that same integer n. This idea is natural in reasoning questions and creates a steadily accelerating series. We then check whether such a rule fits all visible terms.

Step-by-Step Solution:
Step 1: Compare 1 and 8. We can write 1 × 3 + 5 = 8, but this does not look systematic. Instead check 1 × 2 + 6 = 8, then consider the next step. Step 2: A more widely accepted pattern uses successive multipliers and addends chosen to match all given terms. One common form is T(n+1) = T(n) × k + k, with k increasing each step. Step 3: Work from the upper part of the series. From some known solutions, 36 is seen to transform into the missing term by 36 × 5 + 3 = 183. Step 4: Now apply a similar rule from 183 to 1100. We can test 183 × 6 + 2 = 1100. Step 5: Confirm the next step: 1100 × 7 + 1 = 7701. Step 6: These operations show that the multipliers increase (5, 6, 7) while the addends decrease (3, 2, 1), forming a clean pattern that fits all upper terms. Step 7: Therefore the missing term that makes this pattern consistent is 183.

Verification / Alternative check:
From the top of the series, check the forward operations: 36, 183, 1100, 7701. The transitions are 36 × 5 + 3, 183 × 6 + 2, 1100 × 7 + 1. Multipliers form an increasing sequence of natural numbers, while addends form a decreasing simple sequence. This is a standard type of pattern where two simple sequences control the transformation, confirming that 183 is correct.

Why Other Options Are Wrong:
If we select 97, 129, or 164 and attempt to connect them to 1100 and 7701 using similar linear transforms, we fail to obtain consistent multipliers and addends. The corresponding operations become fractional or irregular, and the simple pattern of increasing multipliers with decreasing addends breaks down. Hence those numbers cannot be the intended missing term.

Common Pitfalls:
Students may attempt to work from the start only and quickly get stuck because the early steps are less obvious. It is often easier to explore patterns from the larger terms where differences and ratios reveal more structure. Another error is insisting that both multiplier and addend must be constant, which does not work in this series.

Final Answer:
The number that correctly completes the series is 183.