Number Series Problem Analysis

The question presents a number series: 2, 4, 10, 28, ?, 244. We need to find the number that should replace the question mark (?) based on the pattern of the series.

Let's examine the relationship between consecutive terms to identify the underlying rule governing this number series.

Identifying the Pattern in the Number Series

We can look at the differences between successive terms:

• Difference between the 2nd and 1st term: $4 - 2 = 2$
• Difference between the 3rd and 2nd term: $10 - 4 = 6$
• Difference between the 4th and 3rd term: $28 - 10 = 18$

The differences are 2, 6, 18. Notice that each difference is 3 times the previous difference ($6 = 3 \times 2$, $18 = 3 \times 6$). This suggests a pattern where the increase between terms is growing by a factor of 3.

Let's try to find a direct formula for the terms. Consider the terms: 2, 4, 10, 28, 244.

Let's examine the structure of the terms relative to powers of 3:

• $2 = 1 + 1 = 3^0 + 1$
• $4 = 3 + 1 = 3^1 + 1$
• $10 = 9 + 1 = 3^2 + 1$
• $28 = 27 + 1 = 3^3 + 1$

This reveals a clear pattern: the nth term ($T_n$) in the series appears to be given by the formula $T_n = 3^{n-1} + 1$. Let's verify this formula for the given terms.

Applying the Pattern to Find the Missing Number

Using the pattern $T_n = 3^{n-1} + 1$, let's calculate each term:

• For $n=1$: $T_1 = 3^{1-1} + 1 = 3^0 + 1 = 1 + 1 = 2$. Matches the first term.
• For $n=2$: $T_2 = 3^{2-1} + 1 = 3^1 + 1 = 3 + 1 = 4$. Matches the second term.
• For $n=3$: $T_3 = 3^{3-1} + 1 = 3^2 + 1 = 9 + 1 = 10$. Matches the third term.
• For $n=4$: $T_4 = 3^{4-1} + 1 = 3^3 + 1 = 27 + 1 = 28$. Matches the fourth term.
• For $n=5$: This is the position of the question mark. $T_5 = 3^{5-1} + 1 = 3^4 + 1 = 81 + 1 = 82$.
• For $n=6$: This is the last term. $T_6 = 3^{6-1} + 1 = 3^5 + 1 = 243 + 1 = 244$. Matches the sixth term.

The pattern $T_n = 3^{n-1} + 1$ perfectly fits all the given terms in the series.

The Missing Number in the Series

Based on the pattern, the number that replaces the question mark (?) at the 5th position ($n=5$) is 82.

Number Series Revision Table

Understanding number series patterns is a key skill in logical reasoning. Different types of patterns include arithmetic progression, geometric progression, differences, double differences, squares, cubes, alternating patterns, and combinations of operations.

Additional Information on Number Series

Number series questions test your ability to identify mathematical patterns. Practice with various types of series (like arithmetic, geometric, and mixed series) will help you recognize patterns quickly. Common strategies include looking at differences between terms, ratios between terms, squares/cubes of position numbers or terms, and alternating operations or patterns.