aaron is skipping stones across a pond.(A "skip"is a short bounce in the water.)The first stone skips 1 time, the second skips 2 times, the third stone skips 4 times,and the fourth stone skips 8 times.
if this patten continues,how many total skips will the 10th stone make? how many total skips will have been made?
what is the patten?if yo knew the number of the stone,such as 11th,or n,how would you figure out how many skips it would take?
To find the pattern in the number of skips made by each stone, we can observe that the number of skips doubles with each subsequent stone. From the information given, we can see that the first stone skips 1 time, the second stone skips 2 times, the third stone skips 4 times, and the fourth stone skips 8 times. This pattern continues, where each stone skips twice as many times as the previous stone.
To find the number of skips the 10th stone will make, we can use the pattern. Since each stone doubles the number of skips, we can calculate it by taking 2 to the power of (10-1). So, the 10th stone will make 2^(10-1) = 2^9 = 512 skips.
To find the total number of skips made by all the stones so far, we add up the number of skips of each stone from the first up to the 10th stone. The total number of skips made is the sum of the geometric series: 1 + 2 + 4 + 8 + ... + 2^(10-1).
To calculate the sum of a geometric series, we can use the formula:
Sum = a * (r^n - 1) / (r - 1),
where a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 1 (the first stone skips 1 time), r = 2 (each subsequent stone doubles the number of skips), and n = 10 (we want to find the sum up to the 10th stone).
Plugging these values into the formula, we get:
Sum = 1 * (2^10 - 1) / (2 - 1) = (2^10 - 1) = 1023.
Therefore, the total number of skips made by all the stones so far is 1023.
If we have the number of the stone, such as the 11th stone or the nth stone, we can calculate the number of skips it would take by following the same pattern. Simply calculate 2^(n-1) to find the number of skips for that specific stone.