Question

the cost of renting a kayak for one hour is $23. Each additional hour is 8$ more. Write an explicit formula and recursive formula to represent the situation.

Answer 1

Explicit formula $a_{n} =8n+15$

Recursive formula $\left \{ {{a_{1} =23} \atop {a_{n}=a_{n-1}+8 }} \right.$

To solve this problem we have to use an arithmetic sequence. The cost of renting a kayak for 1 hour is $23, each additional hour is $8 more. So, the first element of the secuence will be 23 for one hour, then each addittional hour will be 23 + 8, making the secuence:

{23, 31, 39, 47, 55,.....,n}

Writing a recursive formula of the form $a_{n} =a_{n-1} +d$ where $a_{n}$ is the nth term, n the number of terms, and d the common difference in the secuence.

The common difference of the secuence {23, 31, 39, 47, 55,.....,n}. So, the first term is $a_{1} =23$, and the common diffenece is d = 8 which is the difference between each term.

$\left \{ {{a_{1} =23} \atop {a_{n} =a_{n-1}+8 }} \right.$

Writing a explicit formula of the form $a_{n} =a_{1} +d(n-1)$ where where $a_{n}$ is the nth term, n the number of terms, $a_{1}$ the first term of the secuence, and d the common difference in the secuence.

With $a_{1}=23$, and d = 8:

$a_{n} =23+8(n-1)$

$a_{n} =23+8n-8$

$a_{n} =8n+15$

Answer 2

X= 23+8y
X represents the total
23 represents the base price
8 represents the price per extra hour
Y represents how many hours extra