John Goodman weighs 250 pounds, so he decides to go on a diet. At the end of the first week, he weighs 240. Then at the end of the second week, he weighs 230.
Fill in the blanks of the arithmetic formula [ A(n) = d + a^1(n – 1) ] for this scenario.
A(n) = ___ + _____ (n – 1 ) How much will he weigh at the end of week 12? 140???__
this is what I got so far
250 + ____ (10 - 1)
idk im so confused
since he loses 10 lbs each week, after n weeks he has lost 10n lbs.
So, starting with week 0 (before he started the diet), he weighed 250 lbs,
so, his weight after n weeks on the diet is 250-10n
However, that doesn't quite set up an arithmetic sequence. The nth term is his weight at the beginning of the nth week, or, his weight at the end of the (n-1)st week.
So, if a1 is his starting weight (after zero weeks),
A(n) = 250 - 10(n-1)
But, an AP is written as An = a1 + (n-1)d, so we need to note that the difference each week is -10 from the previous week. That is,
A(n) = 250 + (n-1)(-10)
To check against our first function, note that
250+(n-1)(-10) = 250 -10n+10 = 260-10n
why not 250-10n? Because the AP starts with his starting weight (week 0 of the diet)
Now just plug in your vales for n to find his weight after n weeks.
To find the values to fill in the blanks of the arithmetic formula and determine John's weight at the end of week 12, let's first analyze the given information.
We know that John's initial weight is 250 pounds. At the end of the first week, he weighs 240 pounds, and at the end of the second week, he weighs 230 pounds.
Let's use the formula A(n) = d + a^(n-1), where:
- A(n) represents the weight at the end of week n,
- d represents the initial weight,
- a represents the common difference (change in weight), and
- n represents the week number.
From week 1 to week 2, John lost 240 - 250 = -10 pounds. Similarly, from week 2 to week 3, he lost 230 - 240 = -10 pounds. We can observe that the weight loss difference is consistent, so the common difference (a) is -10.
Now, let's substitute the known values into the formula to find John's weight at the end of week 12:
A(12) = 250 + (-10)^(12-1)
A(12) = 250 + (-10)^11
A(12) = 250 + (-10,000,000,000)
Note: Since (-10)^11 is a large negative number, John's weight at the end of week 12 would be negative, which is not practically feasible. It seems like the weight loss pattern is not accurately represented in the given scenario.
However, if we assume that John continues losing 10 pounds per week consistently, we can still use the arithmetic formula to find his weight at the end of week 12:
A(12) = 250 + (-10)^(12-1)
A(12) = 250 + (-10)^11
A(12) = 250 + (-10,000,000,000)
A(12) ≈ -9,999,999,750
Therefore, if John continues to lose 10 pounds per week consistently, he would weigh approximately -9,999,999,750 pounds at the end of week 12. However, please note that this result is purely hypothetical and not realistically possible.