Question

1. open response - 2 points

mary is competing in a push up competition and wants to train for it. when she started training, she could do two push ups. last week, she could do 56 push ups, and the week after that she could do 63 push ups. assume this pattern of increase in push ups continues.
a. write the explicit formula to represent this scenario.
b. how many push ups will she do in week 6?

1. selected response - 1 point

Explanation:

Step1: Identify sequence type

This is an arithmetic sequence where the first term $a_1$ is the number of push-ups in week 1, common difference $d$ is the weekly increase.
$a_1 = 12$, $d = 16 - 12 = 4$

Step2: Write explicit formula

The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$. Substitute values:
$a_n = 12 + (n-1) \times 4$
Simplify: $a_n = 12 + 4n - 4 = 4n + 8$

Step3: Calculate week 67 push-ups

Substitute $n=67$ into the explicit formula:
$a_{67} = 4(67) + 8$
$a_{67} = 268 + 8 = 276$

Answer:

a. $\boldsymbol{a_n = 4n + 8}$
b. $\boldsymbol{276}$