Understanding the Number Series

The question asks us to find the next number in the given series: 7, 15, 30, 62, 125, 253, ?

To solve this number series problem, we need to identify the underlying pattern that relates the consecutive terms. Number series can follow various patterns, such as arithmetic progression, geometric progression, or patterns based on differences, sums, products, or a combination of operations.

Analyzing the Differences Between Terms

Let's look at the differences between consecutive terms in the given number series:

• Difference between 15 and 7: ${15 - 7 = 8}$
• Difference between 30 and 15: ${30 - 15 = 15}$
• Difference between 62 and 30: ${62 - 30 = 32}$
• Difference between 125 and 62: ${125 - 62 = 63}$
• Difference between 253 and 125: ${253 - 125 = 128}$

The differences are 8, 15, 32, 63, 128. Let's examine these differences for a pattern.

Identifying the Pattern in Differences

Let the differences be denoted by $d_1, d_2, d_3, d_4, d_5$, where $d_n$ is the difference between the $(n+1)$-th term and the $n$-th term. So we have:

• $d_1 = 8$
• $d_2 = 15$
• $d_3 = 32$
• $d_4 = 63$
• $d_5 = 128$

Let's observe how these differences relate to powers of 2:

• $8 = 2^3$
• $15 = 16 - 1 = 2^4 - 1$
• $32 = 2^5$
• $63 = 64 - 1 = 2^6 - 1$
• $128 = 2^7$

We can see a pattern here. The power of 2 increases sequentially (3, 4, 5, 6, 7). For the first difference ($d_1$, $n=1$), it's $2^{1+2} = 2^3$. For the second difference ($d_2$, $n=2$), it's $2^{2+2} - 1 = 2^4 - 1$. For the third difference ($d_3$, $n=3$), it's $2^{3+2} = 2^5$. For the fourth ($d_4$, $n=4$), it's $2^{4+2} - 1 = 2^6 - 1$. For the fifth ($d_5$, $n=5$), it's $2^{5+2} = 2^7$.

The pattern for the difference $d_n$ seems to be:

• If $n$ is odd, $d_n = 2^{n+2}$
• If $n$ is even, $d_n = 2^{n+2} - 1$

Let's verify this pattern with the given differences:

The pattern holds true for all the calculated differences.

Calculating the Next Term

We need to find the next term in the series, which is the 7th term. This term is obtained by adding the 6th difference ($d_6$) to the 6th term (253).

Using the pattern for $d_n$, for $n=6$ (which is even):

${d_6 = 2^{6+2} - 1}$

${d_6 = 2^8 - 1}$

${d_6 = 256 - 1}$

${d_6 = 255}$

Now, add this difference to the last term of the series:

Next term = Last term + Next difference

Next term = ${253 + 255}$

Next term = ${508}$

Conclusion

Based on the identified pattern in the differences between consecutive terms, the next number in the series 7, 15, 30, 62, 125, 253, ? is 508.

Revision Table: Number Series Pattern

Additional Information on Number Series Patterns

Number series questions test your logical reasoning and pattern recognition skills. Common patterns include:

• Arithmetic Progression: The difference between consecutive terms is constant. Example: 2, 5, 8, 11... (difference is 3)
• Geometric Progression: Each term is obtained by multiplying the previous term by a constant ratio. Example: 3, 6, 12, 24... (ratio is 2)
• Differences/Differences of Differences: Analyzing the sequence of differences between terms, or the differences of those differences, can reveal a pattern.
• Squares and Cubes: Terms might be squares or cubes of numbers, or related to them (e.g., $n^2+1$, $n^3-1$).
• Alternating Patterns: Different rules might apply to alternate terms or different parts of the series.
• Combined Operations: A combination of operations (like multiply and then add/subtract) might be used to get the next term. This is what we observed in this problem's differences pattern ($2^{n+2}$ vs $2^{n+2}-1$). Another way to look at the original series relationship could be $a_n \approx 2 \times a_{n-1}$. Let's quickly check: $2 \times 7 = 14$ (15), $2 \times 15 = 30$ (30), $2 \times 30 = 60$ (62), $2 \times 62 = 124$ (125), $2 \times 125 = 250$ (253). The relationship is roughly doubling, with small adjustments (+$1, +0, +2, +1, +3$). While this also hints at a pattern, the difference pattern we used was more direct for calculation.

Practicing various types of series helps in quickly identifying the pattern during exams.