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

Answer:

$p_n=2^n$

$p_n=32768$

Step-by-step 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.