Directions (61-65) : Find the wrong number in following number series.

5 , 7 , 17 , 47 , 115 , 235, 467

The correct answer is 235.

Step-by-step Explanation:

Let us analyze the pattern followed by the given number series by finding the differences between consecutive terms:

$7-5=2$

$17-7=10$

$47-17=30$

$115-47=68$

Notice that these differences follow the mathematical pattern of $n^3+n$ for consecutive natural numbers $n=1,2,3,4,\dots$:

1³ + 1 = 1 + 1 = 2
2³ + 2 = 8 + 2 = 10
3³ + 3 = 27 + 3 = 30
4³ + 4 = 64 + 4 = 68

Following this pattern, the next two differences should be:

5³ + 5 = 125 + 5 = 130
6³ + 6 = 216 + 6 = 222

Now, let us calculate the terms of the series using these differences:

1st term = 5
2nd term = 5 + 2 = 7
3rd term = 7 + 10 = 17
4th term = 17 + 30 = 47
5th term = 47 + 68 = 115
6th term = 115 + 130 = 245 (given as 235)
7th term = 245 + 222 = 467

Hence, the wrong number in the given series is 235, which should be replaced by 245.