Question

1+1/2+1/4+1/8+1/16+1/32+1/64

Answer 1

Answer: You will never reach a sum of 2. You would need infinitely many terms to sum up.

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Explanation:

We have this sequence

1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...

which is geometric with the following properties

• a = first term = 1
• r = common ratio = 1/2 = 0.5

Notice how we multiply each term by 1/2 to get the next term. Eg: (1/4)*(1/2) = 1/8 or (1/16)*(1/2) = 1/32.

Since r = 0.5 is between -1 and 1, i.e. -1 < r < 1 is true, this means that adding infinite terms of this form will get us to approach some finite sum which we'll call S. This is because the new terms added on get smaller and smaller.

That infinite sum is

S = a/(1-r)

S = 1/(1-0.5)

S = 1/0.5

S = 2

So if we keep going with that pattern 1+1/2+1/4+... and do so forever, then we'll reach a sum of 2. However, we cannot go on forever since it's asking when we'll reach that specific sum. In other words, your teacher wants finitely many terms to be added.

In short, we'll never actually reach the sum 2 itself. We'll just get closer and closer.

Here's a list of partial sums

• 1+1/2 = 1.5
• 1+1/2+1/4 = 1.75
• 1+1/2+1/4+1/8 = 1.875
• 1+1/2+1/4+1/8+1/16 = 1.9375
• 1+1/2+1/4+1/8+1/16+1/32 = 1.96875
• 1+1/2+1/4+1/8+1/16+1/32+1/64 = 1.984375
• 1+1/2+1/4+1/8+1/16+1/32+1/64+1/128 = 1.9921875
• 1+1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256 = 1.99609375

We can see that we're getting closer to 2, but we'll never actually get there. We'd need to add infinitely many terms to get to exactly 2.