Jeff took half of the marbles out of the bag. He then put 1 marble back into the bag. He repeated this process five times. There were 3 marbles left in the bag at the end. How many marbles were there in the bag at first?

work backwards

x5/2 + 1 = 3 so x = 4
x4/2 + 1 = 4 so x4 = 6
work it 3 more times to find x1, the starting amount

Let's work through this problem step by step.

Step 1: At the beginning, we assume there were "x" marbles in the bag.

Step 2: Jeff took half of the marbles out of the bag, which means he removed (1/2)*x marbles from the bag.

Step 3: Jeff put 1 marble back into the bag, so now the number of marbles in the bag is [(1/2)*x] + 1.

Step 4: Jeff repeated this process five times. Each time, he took half of the marbles out and then put 1 marble back in the bag. So, we can represent the number of marbles in the bag after the repeated process as:

[(1/2)*[(1/2)*[(1/2)*[(1/2)*[(1/2)*x + 1] + 1] + 1] + 1] + 1]

Step 5: At the end of the process, there were 3 marbles left, so we can set up the equation as:

[(1/2)*[(1/2)*[(1/2)*[(1/2)*[(1/2)*x + 1] + 1] + 1] + 1] = 3

Simplifying the equation, we get:

[(1/2)*[(1/2)*[(1/2)*[(1/2)*[(1/2)*x + 1] + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/2)*[(1/2)*[(1/2)*x + 2] + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/2)*[(1/4)*x + 1] + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/2)*[(1/4)*x + 2] + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/2)*[(1/4)*x + 3] + 1] + 1] + 1] = 3

Step 6: Now, let's simplify the equation further:

[(1/2)*[(1/2)*[(1/2)*[(1/4)*x + 3] + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/2)*(1/4)*x + (1/2)*3 + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/8)*x + (3/2) + 1] + 1] + 1] = 3
[(1/2)*[(1/2)*[(1/8)*x + 5/2] + 1] + 1] = 3
[(1/2)*[(1/2)*(1/8)*x + (1/2)*(5/2) + 1] + 1] = 3
[(1/2)*[(1/16)*x + (5/4) + 1] + 1] = 3
[(1/2)*(1/16)*x + (1/2)*(9/4) + 1] + 1 = 3
(1/32)*x + (9/8) + 1 + 1 = 3
(1/32)*x + (9/8) + 2 = 3
(1/32)*x + (9/8) + (16/8) = 3
(1/32)*x + (25/8) = 3

Step 7: To remove the fraction, we multiply the equation by 32:

32 * [(1/32)*x + (25/8)] = 32 * 3
x + 25 = 96

Step 8: Subtract 25 from both sides to isolate x:

x = 96 - 25
x = 71

Therefore, there were 71 marbles in the bag initially.

Let's work through the problem step by step to find the answer.

First, let's assume the number of marbles in the bag at the beginning is "x".
Jeff took half of the marbles out of the bag, which means he took (1/2)x marbles.
After taking half out, there are (x - (1/2)x) = (1/2)x marbles left.

Then, Jeff put 1 marble back into the bag, so the number of marbles in the bag at this point is (1/2)x + 1.

He repeated this process five times, so we need to repeat the steps above five times:

After the first repetition: The number of marbles in the bag is (1/2)((1/2)x + 1) + 1.
After the second repetition: The number of marbles in the bag is (1/2)((1/2)((1/2)x + 1) + 1) + 1.
After the third repetition: The number of marbles in the bag is (1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) + 1.
After the fourth repetition: The number of marbles in the bag is (1/2)((1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) + 1) + 1.
After the fifth repetition: The number of marbles in the bag is (1/2)((1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) + 1) + 1.

We know that there are 3 marbles left in the bag at the end, so we can set up the equation:

(1/2)((1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) + 1) + 1 = 3.

Now, we can solve this equation to find the value of x:

(1/2)((1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) + 1) = 2.

(1/2)((1/2)((1/2)((1/2)x + 1) + 1) + 1) = 4.

(1/2)((1/2)((1/2)x + 1) + 1) = 8.

(1/2)((1/2)x + 1) + 1 = 16.

(1/2)x + 1 + 1 = 32.

(1/2)x + 2 = 32.

(1/2)x = 30.

x = 60.

Therefore, there were 60 marbles in the bag at first.