Which of the following numbers will replace the question marks (?) in the given series?
2, ?, ?, 17, 26, 37, 50, 65, 82

The correct option is 5, 10.

To find the numbers that replace the question marks in the series, let us analyze the pattern of differences between consecutive terms.

Given series: 2, ?, ?, 17, 26, 37, 50, 65, 82

Let us check the differences between the known consecutive terms toward the end of the series:

$82-65=17$

$65-50=15$

$50-37=13$

$37-26=11$

$26-17=9$

Notice that the differences between successive terms are consecutive odd numbers: 9, 11, 13, 15, 17.

Extending this pattern backward, the differences preceding 9 should be consecutive smaller odd numbers, which are 3, 5, and 7.

Now, let us calculate the missing numbers step-by-step using these differences:

First missing number (2^nd term):
$2+3=5$

Second missing number (3^rd term):
$5+5=10$

We can verify our result by checking the next term (4^th term):
$10+7=17$
This matches the 4^th term in the given series.

Alternatively, each term in the series can be represented as $n^2+1$ for $n=1,2,3,4,...$:

1^st term: $1^2+1=2$

2^nd term: $2^2+1=5$

3^rd term: $3^2+1=10$

4^th term: $4^2+1=17$

Therefore, the missing numbers are 5 and 10.