Question

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.

Answer 1

Let
`p_n` be the amount of pennies received for lawn
`n`. Then,
`p_1 = 2` and
`p_(n+1) = 2p_n`.

We claim by induction that
`p_n = 2^n`. The base case is trivial (
`p_1 = 2^1 = 2` is given). Then, we complete the inductive step. If
`p^n=2^n`, we have:


`p_(n+1) = 2 \cdot 2^n = 2^(n+1)`

This completes the proof.

Thus,
`p_(15) = 2^(15) = 32768 = $327.68`.

Answer 2

`p_n=2^n`

`p_n=32768`

Explanation:

We are given that your friend asks you to help cut the grass this summer and will pay you 2 pennies for the first job

We are given that you are agreed to help if doubles your payment for each job completed.

After 2 lawns, you will receive money= 4 pennies

After 3 lawns, you will receive money=8 pennies

Let total earn pennies are
`p_n` for lawn n and
`p_1`be the number of pennies receive after first job completed.

`p_1=2`

`p_(n+1)=2p_n`

We have to prove that

`p_n=2^n`

It is proved by induction method

`p_1=2^1=2`

Hence,
`p_1` is true for n=1

Let
`p_k=2^k` is true for n=k

Now, we shall prove that for n=k+1
`p_(n+1)=2^(n+1)` is true

Substitute n=k+1 then we get

`p_(k+1)=2^(k+1)=2\cdot2^k=2p_k`

Hence, it is true for n=k+1

Hence, proved.

`p_n=2^n`

Now substitute n=15 then we get

`p_(15)=2^(15)`

`p_n=32768`

Hence, the number of pennies he will pay you after cutting the 15th lawn=32768.