Question

ian is in training for a national hot dog eating contest on the 4th of july. on his first day of training, he eats 4 hot dogs. each day, he plans to eat 110% of the number of hot dogs he ate the previous day. write an explicit formula that could be used to find the number of hot dogs Ian will eat on any particular day.

Answer 1

The number of hot dogs on any particular day is given by:

`D_n=(1.1)^(n-1)* 4` where n represents the nth day.

Explanation:

Let
`D_n` represents the number of hot dogs in nth day of his training.

As it is given that he eats 4 hot dogs on his first day of training that means:

`D_1=4`

Now it is also given that:

he plans to eat 110% of the number of hot dogs he ate the previous day.

i.e. the recurrence relation is given as:

`D_n=110\%* D_(n-1)`

which could also be written as:

`D_n=1.1* D_(n-1)`

Now:

`D_2=1.1* D_1=1.1* 4\\\\D_3=1.1* D_2=1.1* 1.1* 4=(1.1)^2* 4\\\\D_4=1.1* D_3=(1.1)^3* 4\\\\.\\.\\.\\.\\.\\.\\D_n=(1.1)^(n-1)* 4`

Hence, the number of hot dogs on any particular day is given by:

`D_n=(1.1)^(n-1)* 4` where n represents the nth day.

Answer 2

Answer:
`a_n = 4(1.1)^(n-1)`

Explanation:

Since, the initial number of hot dogs = 4

According to the question,

The number of hot dogs is increasing by 110% of that of previous day,

Thus, the number of hot dog in first day = 4

Second day = 110 % of 4 = 4.4

Third day = 110% of 4.4 = 4.84

Fourth day = 110% of 4.84 = 5.324

So on.......

Thus, we get a GP,

4, 4.4, 4.84, 5.324 ..........................

That having common ratio, d = 1.1

And, first term, a = 4

Since, the nth term of the GP,
`a_n = a* d^(n-1)`

Hence, the required explicit formula of the given situation,

`a_n = 4(1.1)^(n-1)`