Your friend asks you to help cut grass this summer and will pay you 2 pennies for the first job. You agree to help if he doubles your payment for each job completed. After 2 lawns, you will receive 4 pennies, and after 3 lawns, you will receive 8 pennies.
Complete and solve the equation that finds the number of pennies he will pay you after cutting the 15th lawn.
Let's assume that the number of pennies paid after cutting the 15th lawn is represented by P.
From the given information, we know that after the first lawn, you are paid 2 pennies. After the second lawn, you received double that amount, which is 4 pennies. After the third lawn, you received double that amount again, which is 8 pennies. This pattern continues, with the payment doubling each time.
Using this pattern, we can express the number of pennies paid after each lawn as a geometric sequence. The general term of a geometric sequence is given by the formula:
an = a * r^(n-1)
where:
an is the nth term
a is the first term
r is the common ratio
n is the number of terms
In this case, the first term (a) is 2 pennies, and the common ratio (r) is 2 because the payment doubles each time.
So, the equation to find the number of pennies paid after the nth lawn is:
P = 2 * 2^(n-1)
To find the number of pennies paid after the 15th lawn, we substitute n = 15 into the equation:
P = 2 * 2^(15-1)
P = 2 * 2^14
P = 2 * 16,384
P = 32,768 pennies