5, 45, 83, 117, 145, ?

Options

A. 165
B. 177
C. 160
D. 158

Explanation

To find the missing number in the series 5, 45, 83, 117, 145, we will follow these steps:

### Step 1: Identify the series
The given series is:
5, 45, 83, 117, 145

### Step 2: Calculate the differences between consecutive terms
We will find the difference between each pair of consecutive numbers in the series.

- Difference between 45 and 5:
\( 45 - 5 = 40 \)

- Difference between 83 and 45:
\( 83 - 45 = 38 \)

- Difference between 117 and 83:
\( 117 - 83 = 34 \)

- Difference between 145 and 117:
\( 145 - 117 = 28 \)

Now we have the differences:
40, 38, 34, 28

### Step 3: Calculate the differences of the differences
Next, we will find the differences between these differences.

- Difference between 38 and 40:
\( 38 - 40 = -2 \)

- Difference between 34 and 38:
\( 34 - 38 = -4 \)

- Difference between 28 and 34:
\( 28 - 34 = -6 \)

Now we have the second differences:
-2, -4, -6

### Step 4: Identify the pattern in the second differences
The second differences are decreasing by 2 each time:
-2, -4, -6

Following this pattern, the next difference should be:
-6 - 2 = -8

### Step 5: Calculate the next difference
Now we will add this new difference to the last difference we calculated (28):
28 + (-8) = 20

### Step 6: Calculate the next term in the original series
Now we will add this new difference (20) to the last term in the original series (145):
145 + 20 = 165

### Conclusion
The missing number in the series is **165**.

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