In the series 50, 51, 47, 56, 40, ?, find the next number by observing the pattern of adding and subtracting consecutive square numbers.

Introduction / Context:
This number series shows alternating increases and decreases: 50, 51, 47, 56, 40, ?. The jumps are not constant, which suggests that the differences themselves follow a special sequence. A common pattern in such questions is the use of perfect squares as the amounts added or subtracted.

Given Data / Assumptions:
The terms are:

• 50
• 51
• 47
• 56
• 40
• ?

We assume that from one step to the next the series alternately adds and subtracts squared numbers like 1^2, 2^2, 3^2 and so on, in order.

Concept / Approach:
The idea is to calculate the difference between each pair of consecutive terms and see if these differences correspond to square numbers, possibly with alternating signs. If the pattern matches 1^2, 2^2, 3^2, 4^2, 5^2, then we can easily compute the missing term using the next square in the pattern.

Step-by-Step Solution:
Step 1: From 50 to 51, the difference is +1, which is 1^2.Step 2: From 51 to 47, the difference is -4, which is -2^2.Step 3: From 47 to 56, the difference is +9, which is 3^2.Step 4: From 56 to 40, the difference is -16, which is -4^2.Step 5: This shows a pattern: add 1^2, subtract 2^2, add 3^2, subtract 4^2, and so on.Step 6: The next step should follow the pattern and add 5^2 = 25 to the current term 40.Step 7: Therefore, the missing term is 40 + 25 = 65.

Verification / Alternative check:
We can summarise the rule as: starting from 50, repeatedly add or subtract n^2 with signs alternating plus, minus, plus, minus, plus, and so on. That is, 50 + 1 = 51, 51 - 4 = 47, 47 + 9 = 56, 56 - 16 = 40, and 40 + 25 = 65. This reconstruction reproduces all the known terms and produces a unique, consistent value for the missing term.

Why Other Options Are Wrong:
Values 58, 76 and 81 do not equal 40 + 25, and using any of them would break the smooth pattern of adding and subtracting consecutive squares. They would lead to irregular differences that are not square numbers, so they are not supported by the data.

Common Pitfalls:
Students sometimes look only for arithmetic progressions and miss that the differences themselves can form a familiar sequence like 1, 4, 9, 16, 25. Another pitfall is to forget the alternating signs and try to fit a single increasing or decreasing set of differences, which does not work here.

Final Answer:
The next number in the series, obtained by adding 5^2 to 40, is 65.