Question

The chart below shows the number of new clients an insurance agent added each month.

a. If the pattern continues, how many new clients will she add in December?
b. How many new clients will she add in the entire year?

Please show all steps!! I know how to figure this in my head but I need to show work and I don't know how to write the work out for this.

Will post table in comments.

Answer 1

@lonnie455rich

Answer 2

let n be new clients and x be next months amount
n+2=x
12(n+2)<-would be a year.
im sure there is a better way of doing this though. without having to explain the variables. are you doing functions now?.

Answer 3

maybe @bibby will have a better idea

Answer 4

We are doing series and sequences now.

Answer 5

in calc 2?

Answer 6

I understand what youre saying about the year tho. And I'm in Alg 2b

Answer 7

ahh.. hmmm.. there has got to be a better way of doing it. maybe your just supposed to show your work with numbers?
because if theyre going up by 2 step a should just be that.

Answer 8

and it wouldnt be what i put for the year. it would be 2*12

Answer 9

because there are 12 months and she is adding 2 per month

Answer 10

i was getting to proofy for alg2 and messed it all up.

Answer 11

Well this problem is from a "Find the Errors" assignment. We get questions and get to see someone else's work on the problem. We have to solve the problem and then explain what the other persons mistake was. I can post it if you wanna see it.

Answer 12

sure

Answer 13

Can you see it?

Answer 14

ya. im going to have to think about this for a second. although there wont be 32 new clients in december. should be 27

Answer 15

they did order of operations wrong

Answer 16

where they have 5+11*2 they did it 16*2=32 instead of 5+22=27

Answer 17

Oh okay I see. I was able to figure out the 27 in my head but i didnt know how to set it into an equation to prove my work.

Answer 18

ya i was trying to find a mistake in the algorithm not the math. making my head spin

Answer 19

on the second one they used the 32 from part a instead of 27. i would confirm that by getting the sum of all the months to see if it adds up correctly

Answer 20

a)

In January, we start with 5 clients.

In February, we now have 7 clients. We've added 2 clients.

In March, we now have 9 clients. We've added 2 clients.

In April, we now have 11 clients. We've added 2 clients.

So the pattern is to simply add 2 to each number to get the next term. In May for instance, we'll have 11+2 = 13 clients and in June we'll have 13+2 = 15


Now you could add 2 to each month to get the next month until you reach december, but that's an inefficient way to do it. A much better way is to use the arithmetic sequence formula


a(n) = a1 + d(n-1)


a(n) = 5 + d(n-1) ... first term is a1 = 5


a(n) = 5 + 2(n-1) ... common difference, or the constant added each time, is d = 2


a(n) = 5 + 2n-2


a(n) = 2n + 3


Now we can figure out how many clients we'll have for any given month and we don't have to know the previous month.


Example: in march, n = 3, so


a(n) = 2n + 3


a(3) = 2(3) + 3


a(3) = 9


And this matches up with what the table says

Answer 21

So what you need to do to finish up part A is to plug in n = 12 (since December is the 12th month) and evaluate to figure out how many new clients are added in December.

Answer 22

So it would be a12=27? I think that's what Lonnie said before too.

Answer 23

That is correct. So that's the answer to part a)

Answer 24

well, in the problem the algorithm was given and you were supposed to find the mistake. that is where i got confused. the mistake was the problem misused order of operations. then used the wrong number for december for part b and messed that one up.

Answer 25

So how would I go about finding the total number of clients?

Answer 26

By using the formula

\[\Large S = \frac{n(a_1+a_n)}{2}\]

this will give you a handy shortcut to adding up the first n terms of a sequence.

Answer 27

its given in part b. the error is that they used 32 instead of 27

Answer 28

So would it be 192 new clients?

Answer 29

that is correct

Answer 30

Ooooh exciting. :) Thanks for your help! Both of you :)

Answer 31

you're welcome