Arithmetic sequences, Please Help!

[Joyce]

Hey I have been struggling with 2 problems for awhile even though I don't think it's too difficult.......Any help would be greatly appreciated. The question is;

1)The arithmetic sequence 200, 185, 170,.....is given. What is the first negative term of this sequence?

I have this so far (not sure if i'm on the right track),

a= 200
d=-15 tn=a+(n-1)(-15)

= 200-15n+15



2) A pile of bricks is arranged in rows. The number of bricks in eah row forms the aritmetic sequence; 65, 59, 53,....

a) One row conatins 17 bricks, which row is this?

b) How many bricks are there?

 

[Mrspi]

Hey I have been struggling with 2 problems for awhile even though I don't think it's too difficult.......Any help would be greatly appreciated. The question is;

1)The arithmetic sequence 200, 185, 170,.....is given. What is the first negative term of this sequence?

I have this so far (not sure if i'm on the right track),

a= 200
d=-15 tn=a+(n-1)(-15)

= 200-15n+15



2) A pile of bricks is arranged in rows. The number of bricks in eah row forms the aritmetic sequence; 65, 59, 53,....

a) One row conatins 17 bricks, which row is this?

b) How many bricks are there?

You've made a good start on number 1. Now, you have an expression for the nth term of the sequence, and you want to know when this is FIRST going to be a negative number (less than 0):
215 - 15n < 0
215 < 15n
215/15 < n
The first integral value of n which is greater than 215/15 is 15, so the 15th term of the sequence will be the first one which is negative.

You can approach problem 2 the same way. Start by writing an expression for the nth term (a = 65 and d = -6)....then set the expression equal to 17 and solve for n. That will tell you the number of the row which contains 17 bricks.

I'm not sure about the "how many bricks" question....do they mean how many bricks in the rows up to and including the one with the 17 bricks? If so, use the formula for the sum of the first n terms of the sequence:

S<SUB>n</SUB> = (n/2)(a<SUB>1</SUB> + a<SUB>n</SUB>)

You know a<SUB>1</SUB> and a<SUB>n</SUB>, and you will have found the value of n in the first part of the problem.

I hope this helps you.

 

[Joyce]

o.k I understand now thanks alot

 

[soroban]

Hello, Joyce!

2) A pile of bricks is arranged in rows.
The number of bricks in each row forms the aritmetic sequence; 65, 59, 53,....

a) One row conatins 17 bricks, which row is this?

b) How many bricks are there?

We have an arithmetic sequence with first term \(\displaystyle a_1\,=\,65\) and common difference \(\displaystyle d\,=\,-6\)

The n^th term of an arithmetic sequence is: \(\displaystyle \,a_n\:=\:a_1\,+\,d(n\,-\,1)\)

If the n^th term is 17, we have: \(\displaystyle \:65\,-\,6(n\,-\,1)\:=\:17 \;\;\Rightarrow\;\;n\,=\,9\;\) (a)


Assuming that we cannot have "negative bricks", we must stop at the 11th row.

\(\displaystyle \;\;\)The n^th row contains: \(\displaystyle a_1\,+\,d(n\,-\,1)\) bricks.

\(\displaystyle \;\;\)Since the number of bricks in a row must be nonnegative: \(\displaystyle \,65\,-\,6(n\,-\,1)\:\,\geq\,0\)

\(\displaystyle \;\;\)Solve for \(\displaystyle n\) and we get: \(\displaystyle \:n\,\leq\,\frac{71}{6}\;\;\Rightarrow\;\;n\,\leq\,11\)


The sum of the first \(\displaystyle n\) terms of the sequence is: \(\displaystyle \:S_n\:=\:\frac{n}{2}\left[2\cdot a_1\,+\,d(n\,-\,1)\right]\)

The eleven rows contain: \(\displaystyle \:S_{11}\;=\;\frac{11}{2}\left[2\cdot65\,-\,6(10)\right] \;=\;385\) bricks.