Question

Find the missing number: 5, 7, 13, 25, 45, ________

Hint:

Look at the differences between consecutive terms as a set of numbers on their own: 2, 6, 12, 20. Try writing each one as a product of two consecutive whole numbers instead of subtracting them again to find a second-level pattern, and see what product comes next in that same pattern.

Answer 1

Step 1: Find the differences between consecutive terms.
\[ 7 - 5 = 2,\quad 13 - 7 = 6,\quad 25 - 13 = 12,\quad 45 - 25 = 20 \]
Step 2: Analyze the pattern of differences.
The differences are:
\[ 2,\; 6,\; 12,\; 20 \]
These increase by:
\[ +4,\; +6,\; +8 \]
Step 3: Find the next difference.
Following the pattern, the next increase will be \( +10 \), so:
\[ 20 + 10 = 30 \]
Step 4: Find the missing number.
\[ 45 + 30 = 75 \]
Step 5: Conclusion.
The missing number in the series is 75.

Answer 2

Concept:

• Instead of finding first and second differences step by step, the differences between consecutive terms can sometimes be recognised directly as products of two consecutive whole numbers, which predicts every future difference at once.
• Recognising the pattern in one step, rather than building a difference table, avoids the risk of a small arithmetic slip while extending the table.

Step 1: List the differences between consecutive terms.
$7-5=2,\quad 13-7=6,\quad 25-13=12,\quad 45-25=20$.

Step 2: Rewrite each difference as a product of two consecutive whole numbers.
$2 = 1\times2,\quad 6 = 2\times3,\quad 12 = 3\times4,\quad 20 = 4\times5$.

Step 3: Use the pattern to predict the next difference directly.
The next product of consecutive whole numbers in this sequence is $5\times6=30$, so the next difference must be $30$.

Step 4: Add the predicted difference to the last given term.
$45 + 30 = 75$.

Final Answer: The missing number is 75.