Explanation:

Let’s break it down:

- Here’s the sequence: 3, 20, 38, 52, 75, 82, 114, ?

- See the jumps between numbers:

- 20–3 = 17

- 38–20 = 18

- 52–38 = 14

- 75–52 = 23

- 82–75 = 7

- 114–82 = 32

- Those differences aren’t following any simple, repeating addition. Time to look a bit deeper.

Patterns in the gaps:

- If you check, the gap alternates between increasing and then a random small one—let’s jot them out: 17, 18, 14, 23, 7, 32.

- Weird, right? But if you sum every second gap, for example:

- Starting at 17, skip one: 17 + 14 + 7 = 38

- Next, 18 + 23 + 32 = 73

- But that's not tracking with the actual numbers.

Hold on, what if you consider they’re sometimes adding, sometimes subtracting a fixed or variable value? Let's see:

- Between 52 and 75 is a jump of 23—then suddenly just 7, then a big 32. That's an inconsistent pattern.

Let’s try plugging in options to the last known number, 114:

- Option 1: 104 — this would be a drop, which makes no sense given the last jump was up.

- Option 2: 101 — much the same, unlikely.

- Option 3: 118 — this is an increase of only 4, which seems out of character, given the last jump was +32.

- Option 4: 110 — increase of -4. That’d *decrease* the number, which doesn’t fit the upward pattern.

So, let’s check which fits the logic best. From 114, the numbers before have followed 7, 32—maybe the next jump is even bigger? But option 118 (a +4 jump) doesn’t match. The only one that doesn’t break the sequence’s upward momentum would be option 3: 118.

Correct answer: Option 3—118.

Here’s why:

- The last difference is 32, before that it was 7, then 23. The jumps spike sharply and then dip. The pattern isn’t simple, but the only number to keep the direction going up, without a weird dip, is 118.

- All other options would oddly decrease or barely change the number, which would break the visible upward trend.

So, the answer’s 118, not 110.