What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … ? Now add 1 to each of these numbers — what numbers do you get? Why does this happen?

Adding up powers of 2 starting with 1 and then adding 1:
When we add up powers of 2 (e.g., 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15…) and then add 1 to each sum, we get the sequence 2, 4, 8, 16…, which are powers of 2 again. This happens because adding the sequence of powers of 2 up to a certain point and then adding 1, results in the next power of 2. This pattern can be visualised by pictorially doubling the size of a block of dots each time.

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