Your friend asks you to help him with his dog grooming business and will pay you 4 pennies for the first job. You agree to help if he quadruples your payment for each job completed. After 2 grooming jobs, you will receive 16 pennies, and after 3 grooming jobs, you will receive 64 pennies.
Complete and solve the equation that finds the number of pennies he will pay you after the 9th dog grooming job.
Payment for the 9th grooming job = __________ pennies
To find the payment for the 9th dog grooming job, we start with the pattern you’ve given. It appears that the payment quadruples (increases by a factor of 4) with each job.
1st job: 4 pennies
2nd job: 4 * 4 = 16 pennies
3rd job: 16 * 4 = 64 pennies
From this pattern, we can observe that the payment on job ( n ) can be expressed with the formula:
[ P(n) = 4 times 4^{(n-1)} ]
Now, let’s calculate for the 9th job:
[ P(9) = 4 times 4^{(9-1)} ]
[ P(9) = 4 times 4^{8} ]
[ P(9) = 4 times 65536 ]
[ P(9) = 262144 ]
So, the payment for the 9th grooming job is 262,144 pennies.