Number Series Analysis

We are asked to find the missing number in the given number series: 35, 36, 40, ?, 83, 208, 244. The goal is to identify the underlying mathematical pattern.

Sequence Progression Steps

First, let's calculate the differences between consecutive terms to see if there's an obvious arithmetic progression:

• The difference between the second term (36) and the first term (35) is $36 - 35 = 1$.
• The difference between the third term (40) and the second term (36) is $40 - 36 = 4$.
• Let the missing fourth term be represented by '$x$'. The difference between the fourth term and the third term is $x - 40$.
• The difference between the fifth term (83) and the fourth term ($x$) is $83 - x$.
• The difference between the sixth term (208) and the fifth term (83) is $208 - 83 = 125$.
• The difference between the seventh term (244) and the sixth term (208) is $244 - 208 = 36$.

The sequence of differences is: 1, 4, $(x - 40)$, $(83 - x)$, 125, 36.

Identifying the Pattern in Differences

Now, let's examine the sequence of differences: 1, 4, ?, ?, 125, 36. We need to find a pattern that relates these differences.

Let's consider the possibility that the amount added relates to the position of the term in the sequence. Let the terms be $T_1, T_2, T_3, T_4, T_5, T_6, T_7$. The operation happens between $T_n$ and $T_{n+1}$.

• $T_1 = 35$
• $T_2 = 36 = 35 + 1$. The added value is $1$. We can write $1$ as $1^2$. (Here $n=1$)
• $T_3 = 40 = 36 + 4$. The added value is $4$. We can write $4$ as $2^2$. (Here $n=2$)
• Let's hypothesize a pattern involving squares ($n^2$) and cubes ($n^3$) of the index '$n$'. Notice the differences 1, 4, 125, 36. These resemble $1^2, 2^2, 5^3, 6^2$.
• Let's test the pattern: Add $n^2$ if $n$ is 1 or $n$ is even. Add $n^3$ if $n$ is odd and $n > 1$.
• For $n=1$: Add $1^2$. $T_2 = T_1 + 1^2 = 35 + 1 = 36$. (Matches)
• For $n=2$: Add $2^2$. $T_3 = T_2 + 2^2 = 36 + 4 = 40$. (Matches)
• For $n=3$: According to the pattern (n is odd and > 1), add $3^3$. $T_4 = T_3 + 3^3 = 40 + 27 = 67$. This is our potential missing number.
• For $n=4$: According to the pattern (n is even), add $4^2$. $T_5 = T_4 + 4^2 = 67 + 16 = 83$. (Matches the given fifth term)
• For $n=5$: According to the pattern (n is odd and > 1), add $5^3$. $T_6 = T_5 + 5^3 = 83 + 125 = 208$. (Matches the given sixth term)
• For $n=6$: According to the pattern (n is even), add $6^2$. $T_7 = T_6 + 6^2 = 208 + 36 = 244$. (Matches the given seventh term)

The pattern holds true for all given terms. The sequence of operations is to add $1^2, 2^2, 3^3, 4^2, 5^3, 6^2$.

Calculating the Missing Value

To find the missing number (the fourth term), we apply the rule for $n=3$:

Missing Term ($T_4$) = $T_3 + 3^3$

Missing Term ($T_4$) = $40 + 27$

Missing Term ($T_4$) = $67$

Thus, the number that replaces the question mark in the series is 67.