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. by the next week, she could do six push ups and the week after that she could do eighteen 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?

Explanation:

Step1: Identify sequence type

The sequence is 2, 6, 18... which is geometric (each term is multiplied by 3). For a geometric sequence, the explicit formula is $a_n = a_1 r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the week number.

Step2: Define values for formula

$a_1 = 2$ (push-ups at start/week 1), $r = 3$ (ratio of consecutive terms: $\frac{6}{2}=3$, $\frac{18}{6}=3$). Substitute into the formula:
$a_n = 2 \times 3^{n-1}$

Step3: Calculate week 6 push-ups

Substitute $n=6$ into the explicit formula:
$a_6 = 2 \times 3^{6-1} = 2 \times 3^5$

Step4: Compute the final value

Calculate $3^5 = 243$, then $2 \times 243 = 486$

Answer:

a. $a_n = 2 \times 3^{n-1}$ (where $n$ = week number, $a_n$ = number of push-ups)
b. 486 push-ups