The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b – 1 is written. What will be the number left on the board at the end?
Explanation:
Initial sum of the terms of the sequence 1, 2, 3, ...., 40 = 40×412 = 820.
When two numbers (a + b) are replaced by (a + b - 1), the total decreases by 1.
After erasing two numbers a and b, and replacing with (a + b − 1), the new sum of the terms of the sequence = 820 − 1
Similarly, after every operation, the sum of the terms of the sequence reduces by 1.
There are a total of 40 numbers and in each step 2 numbers are replaced by 1 number i.e., the number of numbers reduces by 1. For 40 numbers to reduce to just 1 number, this process must be repeated a total of 39 times.
∴ The last number left (i.e. final sum) = 820 − 39 = 781
Hence, option (c).
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