`5,6,14,40,89,170,291`

Answer

To solve the problem of identifying the number that does not fit in the series `5, 6, 14, 40, 89, 170, 291`, we will analyze the pattern in the series step by step.

### Step 1: Identify the pattern
We start by looking for a pattern in the series. The numbers are:
- 5
- 6
- 14
- 40
- 89
- 170
- 291

### Step 2: Analyze the first number
The first number is 5. We can see that:
- \(5 \times 1 + 1^2 = 5 + 1 = 6\)

### Step 3: Analyze the second number
Now, let's analyze the next number:
- \(6 \times 2 + 2^2 = 6 \times 2 + 4 = 12 + 4 = 16\) (This doesn't match the next number, which is 14)

### Step 4: Check the third number
Let’s check the next number:
- \(6 \times 2 + 3^2 = 6 \times 2 + 9 = 12 + 9 = 21\) (This also doesn't match)

### Step 5: Analyze the third number correctly
Instead, let’s analyze the third number directly:
- \(6 \times 2 + 2^2 = 6 + 8 = 14\) (This matches)

### Step 6: Continue the pattern
Now, let’s check the next number:
- \(14 \times 3 + 3^2 = 14 \times 3 + 9 = 42 + 9 = 51\) (Again, this doesn't match)

### Step 7: Check the fourth number
Let’s analyze the fourth number:
- \(14 \times 2 + 4^2 = 14 \times 2 + 16 = 28 + 16 = 44\) (This also doesn't match)

### Step 8: Analyze the fourth number correctly
Instead, let’s analyze the fourth number directly:
- \(14 + 26 = 40\) (This matches)

### Step 9: Continue the pattern
Continuing this way, we can check:
- \(40 + 49 = 89\)
- \(89 + 81 = 170\)
- \(170 + 121 = 291\)

### Step 10: Identify the wrong number
From our analysis, we see that the number `6` does not fit the pattern correctly. The correct number should have been `15` instead of `6`, as it follows the pattern of adding squares.

### Conclusion
Thus, the number that does not fit in the series is `6`, and it should be replaced with `15`.

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