Question 12 of 19intermediate🔧 ApplyShort Answer3 marks

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?

Correct Answer

Sums of powers of 2: 1 = 1 1 + 2 = 3 1 + 2 + 4 = 7 1 + 2 + 4 + 8 = 15 1 + 2 + 4 + 8 + 16 = 31

Pattern: These are all one less than powers of 2: 2¹ - 1 = 1 2² - 1 = 3 2³ - 1 = 7 2⁴ - 1 = 15 2⁵ - 1 = 31

Adding 1 to each: 2, 4, 8, 16, 32, ...

These are the Powers of 2 sequence.

Why this happens: The sum 1 + 2 + 4 + 8 + ... + 2^(n-1) = 2n2^{n} - 1 Adding 1 gives exactly 2n2^{n}. This is a fundamental property of binary numbers and geometric series.

Exercise: Figure it Out (Section 4) | Q: 6 | (Page Page 8)
For More Understanding

Explanation

The sum of powers of 2 from 2⁰=1 to 2^(n-1) equals 2n2^{n} - 1. This is because in binary representation, numbers like 1, 3, 7, 15, 31 are represented as 1, 11, 111, 1111, 11111 in binary, and adding 1 makes them 10, 100, 1000, 10000, 100000 which are powers of 2. This relationship between sums of powers of 2 and powers of 2 themselves is fundamental to understanding binary arithmetic.

Solution Steps

  1. Calculate sums of powers of 2 starting with 1: 1, 3, 7, 15, 31, ...

  2. Observe each sum is one less than a power of 2: 2¹-1, 2²-1, 2³-1, ...

  3. Add 1 to each sum to get: 2, 4, 8, 16, 32, ... (powers of 2)

  4. Apply the formula: 1 + 2 + 4 + ... + 2^(n-1) = 2n2^{n} - 1

  5. Adding 1 to both sides gives exactly 2n2^{n}

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