In the sequence (2,7,17,32,52,\ldots), the successive differences are (5,10,15,20,\ldots). What will be the next term?

(77)

Here the useful pattern is not in the terms themselves but in the differences between consecutive terms. From 2 to 7 the increase is 5, from 7 to 17 it is 10, from 17 to 32 it is 15, and from 32 to 52 it is 20. These differences increase by 5 each time. The next difference must therefore be 25. Adding that difference to the last known term gives 52 plus 25 equals 77, so option C is correct.

The calculation should be performed in two stages: first extend the difference pattern, then apply it to the last term. The difference list is 5, 10, 15, 20, 25, so the next term is 52 + 25. This equals 77. A value of 72 would add only 20 again, while 75 or 80 does not follow the next difference in the stated pattern. The supplied answer C is therefore consistent, and its explanation correctly emphasizes that recognizing the successive differences is the key step.

This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Sequences.