Analyze the Number Series

We are given a number series: 6, 7, 16, 41, ?, ?, 292. Our task is to find the two missing numbers that logically follow the established pattern.

Identify the Series Pattern

Let's denote the terms of the series as $a_1, a_2, a_3, \dots$. So, $a_1 = 6, a_2 = 7, a_3 = 16, a_4 = 41$, and $a_7 = 292$. We need to find $a_5$ and $a_6$.

First, let's look at the differences between consecutive terms:

• $a_2 - a_1 = 7 - 6 = 1$
• $a_3 - a_2 = 16 - 7 = 9$
• $a_4 - a_3 = 41 - 16 = 25$

The sequence of differences is 1, 9, 25. We notice that these are perfect squares:

• $1 = 1^2$
• $9 = 3^2$
• $25 = 5^2$

The pattern appears to be that we add the square of consecutive odd numbers ($1^2, 3^2, 5^2, \dots$) to the previous term to obtain the next term.

Calculate the Missing Terms

Using this pattern, the subsequent differences will be $7^2$ and $9^2$.

Calculating the 5th term ($a_5$):

We add the next square in the sequence, $7^2$, to the 4th term ($a_4 = 41$):

$$ a_5 = a_4 + 7^2 $$

$$ a_5 = 41 + 49 $$

$$ a_5 = 90 $$

So, the first missing number is 90.

Calculating the 6th term ($a_6$):

Next, we add the following square, $9^2$, to the 5th term ($a_5 = 90$):

$$ a_6 = a_5 + 9^2 $$

$$ a_6 = 90 + 81 $$

$$ a_6 = 171 $$

So, the second missing number is 171.

Confirm the Pattern with the Final Term

Let's verify our calculations by checking if the pattern holds for the last term ($a_7 = 292$). The next odd square is $11^2$.

We add $11^2$ to the 6th term ($a_6 = 171$):

$$ a_7 = a_6 + 11^2 $$

$$ a_7 = 171 + 121 $$

$$ a_7 = 292 $$

This matches the given last term of the series, confirming that our calculated missing numbers (90 and 171) are correct.

Final Answer

The numbers that should replace the question marks are 90 and 171.