A series is given with one term missing. Select the correct alternative from the given ones that will complete the series: 80, 130, 190, 260, ?

Introduction / Context:
This series grows in a controlled way where each step increases by a larger amount than the previous step. The differences between terms themselves form an arithmetic progression with a constant increment. Recognising this secondary progression allows us to find the missing term in the series easily.

Given Data / Assumptions:

• Series: 80, 130, 190, 260, ?
• One next term is missing.
• The jump between terms is not constant but increases steadily.

Concept / Approach:
We begin by calculating the differences between consecutive terms. If these form an arithmetic progression, we can find the next difference by adding the common difference, and then add that to the last term to obtain the missing value. This method converts a slightly complex series into a simpler one at the level of first differences.

Step-by-Step Solution:
Step 1: Compute the first differences. 130 - 80 = 50. 190 - 130 = 60. 260 - 190 = 70. Step 2: Collect these differences: 50, 60, 70. Step 3: Observe that the differences themselves form an arithmetic sequence with a common difference of 10. Step 4: The next difference should therefore be 70 + 10 = 80. Step 5: Add this new difference to the last known term of the series: 260 + 80 = 340.

Verification / Alternative check:
Extend the series with the found value: 80, 130, 190, 260, 340. Recalculate differences: 50, 60, 70, 80. These differences clearly increase by 10 each time, which is a very simple and natural pattern. This confirms that 340 is the correct missing term and that the underlying structure of the series is consistent.

Why Other Options Are Wrong:
If we choose 350, 300, or 320, the last difference will not be 80, and the differences will not continue the 50, 60, 70, 80 pattern. For example, 350 would give a final difference of 90, breaking the constant increment of 10 in the difference sequence. Thus these options do not fit the clean arithmetic progression of first differences and must be rejected.

Common Pitfalls:
Some candidates see that the numbers are increasing quickly and try to impose a multiplicative pattern, which does not work here. Others may compute only the first couple of differences and stop too soon. It is important to compute all available differences and look for simple arithmetic progressions among them, especially when the main series shows steadily increasing step sizes.

Final Answer:
The missing number in the series is 340.