As payment for his daughters yard work, a father agrees to give his daughter an allowance of $3.50 in the first week of the year with an increase of 50 cents each week until the last week of the year.
A) How much money did she receive for an allowance in the last week of the year?
B) What was the total amount of money her father gave her in allowances for the year?
I tried using a formula to solve this, and I thought it was the correct one, but I got a totally "way off" answer.
The formula I used was...
Sn = n[2a + (n-1)d]
The other formulas I learned about are:
Sn = n(a+tn)/2
tn = t1 + (n-1)d
I have no idea which one to use and how to solve. Really in need of help.
Your first formula is incorrect , it should say
Sn = (n/2)[2a + (n-1)d ]
for a) you want to know what the value of term52 is
so
term52 = 3.5 + 51(.5)
= 29
you would use your first formula if you knew the first term, the common difference and the number of terms you have.
That is the case in your problem b)
a = 3.5 , d = .5, and n = 52
so total amount of money
= 26[7 + 51(.5)] = 845
the second Sn formula can be used if you know the first and last terms.
Since it is easier to use the second version, and we found the last term in a) we should get the same result
Sum52 = 26( 3.5 + 29) = 845
It should be easy to decide if you looking for the SUM of terms or if you are looking to find a particular TERM
Sorry I forgot part of the formula, it was: Sn = n[2a + (n-1)d]/2
Where did you get 52 from? Did you divide 365 by 7? I got 52. something, didn't get the whole number 52.
mmmhhh.
How many weeks are there in a year again?
1 year = 52.177457 weeks
To answer these questions, we need to understand the given scenario and apply the correct formula. Let's break it down step by step.
Given:
- The first week's allowance is $3.50.
- The allowance increases by 50 cents each week.
- We need to find the allowance for the last week and the total amount for the year.
A) How much money did she receive for an allowance in the last week of the year?
To determine the allowance for the last week, we need to know the number of weeks in a year. There are typically 52 weeks in a year. If we start with $3.50 and the allowance increases by 50 cents each week for 52 weeks, we can calculate the last week's allowance using the formula:
tn = t1 + (n-1)d
where:
tn is the term we want to find (the allowance for the last week)
t1 is the first term (the initial allowance of $3.50)
n is the number of terms (52 weeks)
d is the common difference (50 cents increase per week)
Plugging in these values into the formula:
tn = 3.50 + (52-1) * 0.50
= 3.50 + 51 * 0.50
= 3.50 + 25.50
= $29.00
So, the daughter received $29.00 for an allowance in the last week of the year.
B) What was the total amount of money her father gave her in allowances for the year?
To find the total amount, we need to sum up the allowances for each week. To do this, we can use the following formula:
Sn = n(a + tn)/2
where:
Sn is the sum we want to find (the total amount of the allowances for the year)
n is the number of terms (52 weeks)
a is the first term (the initial allowance of $3.50)
tn is the last term (the allowance for the last week, which we found in part A)
Plugging in these values into the formula:
Sn = 52 * (3.50 + 29.00)/2
= 52 * 32.50/2
= 52 * 16.25
= $845.00
So, the total amount of money her father gave her in allowances for the year is $845.00.
I hope this explanation helps you understand how to solve the problem! Let me know if you have any further questions.