In this number tree, the integers greater than or equal to 0 are written out in increasing order, with the top row containing one integer and every row after containing twice as many integers as the row above it. If row 1 contains the integer 0, what is the fifth number in row 10?

The fifth integer in row 10 using the number tree pattern is 515.

Step-by-step explanation:

The attached picture is instrumental to understanding the solution to this. There are different methods to approach this problem. But we will use observations to solve the problem.

Looking at the tree diagram, we see that the first row has 1 integer, the second has 2, the third has 4 and the fourth has 8. From here, we can see that a row has 2^n numbers where n = 0, 1, 2, 3, 4, . . .

Row 1 = 2^0 = 1 number

Row 2 = 2^1 = 2 numbers

Row 3 = 2^2 = 4 numbers

Row 4 = 2^3 = 8 numbers. And so on.

Also we notice that the last number in row 4 is 14. This can be gotten by adding the total number of numbers from row 1 to row 4 which is

1 + 2 + 4 + 8 = 15 but the first number on the chart is 0 which means we have to take out the first integer to get the last integer on row 4. Doing this, we have 14.

Since our aim is to know the fifth number in row 10, then we need to know the last number in row 9 then count from the first number in row 10 to the fifth. Using 2^n to get the number of numbers in each row like we've done above we can add the number of numbers to know the total numbers we have from row 1 to row 9 as follows:

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 = 511

As we've said previously, 0 is the first integer in the tree and should be removed to know the last number in row 9. This would be 510.

If 510 is the last number in row 9 is 510, then the first number in row 10 would be 511. Counting to the fifth number we get to 515. Thus the fifth number in row 10 using the number tree pattern is 515.