Help!

[Argile1845]

I have a list of five numbers, where the first number is one and the fifth number is five. Each of the other numbers is equal to one more than the average of its two neighbors. What is the sum of all five numbers in the list?

 

[Dr.Peterson]

I have a list of five numbers, where the first number is one and the fifth number is five. Each of the other numbers is equal to one more than the average of its two neighbors. What is the sum of all five numbers in the list?

Here's one way: call the middle number x and write expressions for the second and fourth. Solve for x.

When you follow the guidelines and show us what you've tried, we'll have a much better idea how to help.

 

[Argile1845]

I understand making an equation with x If the numbers were one or two more than their neighbors but it's saying one more than the average... And that's what stumping me. I'm not I'm sure how to go about starting it at all when it's one more than the average. I guess I'm not sure how to write it out to go about figuring it out.

 

[Dr.Peterson]

I understand making an equation with x If the numbers were one or two more than their neighbors but it's saying one more than the average... And that's what stumping me. I'm not I'm sure how to go about starting it at all when it's one more than the average. I guess I'm not sure how to write it out to go about figuring it out.

Suppose two numbers are 1 and x. What expression represents their average? Then add 1 to that.

 

[Argile1845]

Would that be 1 + 1+x/2?

 

[Argile1845]

So what I'm looking at is the number one with variables x y and z in the middle followed by number five.
To find x, I would write the expression x= 1+ (y+1/2)

 

[Deleted member 4993]

Would that be 1 + 1+x/2?

You need to put parentheses to indicate proper order of operation. It should be written as:

average = (1 + 1+x)/2

 

[Dr.Peterson]

So what I'm looking at is the number one with variables x y and z in the middle followed by number five.
To find x, I would write the expression x= 1+ (y+1/2)

The way I did it is a little easier, just using one variable. If x is in the middle, then the second place is (1+x)/2 + 1, and the fourth place is a similar expression. Then x has to be 1 more than the average of those two expressions.

But your approach with three variables can also work, if you have learned how to solve a system of three equations.

 

[Argile1845]

The way I did it is a little easier, just using one variable. If x is in the middle, then the second place is (1+x)/2 + 1, and the fourth place is a similar expression. Then x has to be 1 more than the average of those two expressions.

But your approach with three variables can also work, if you have learned how to solve a system of three equations.

Haha, my approach was not easier. No I have not figured out how to solve the 3 equations yet. Probably why I was getting hung up.

 

[Steven G]

Show us what you got and we'll help you solve your equations for x, y and z.

 

[Argile1845]

This is what I have but it looks wrong to me...
So I would start with number one and move to x then the two after it then get to five and this is what I come up with honestly.
1 + x + ((1+ X )/2 + 1 ) + (5 + (1+X)/2+1) + 5.

 

[Argile1845]

If I go about solving this... Where is the = sign and what is it equal to? Generally it's something like x +1=23 or set up something like this. I'm lost.

 

[Dr.Peterson]

If I go about solving this... Where is the = sign and what is it equal to? Generally it's something like x +1=23 or set up something like this. I'm lost.

Here is what I suggested:

If x is in the middle, then the second place is (1+x)/2 + 1, and the fourth place is a similar expression. Then x has to be 1 more than the average of those two expressions.

You have written an expression for the sum that they ask for, but what you need is an equation that says what I put in bold above.

So, what is the average of the two expressions? Write an equal sign, followed by x, and then solve.

 

[HallsofIvy]

I have a list of five numbers, where the first number is one and the fifth number is five. Each of the other numbers is equal to one more than the average of its two neighbors. What is the sum of all five numbers in the list?

The five numbers are 1, \(\displaystyle x_1\), \(\displaystyle x_2\),\(\displaystyle x_3\), 5. Since each of the numbers is "one more than the average of its neighbors" we have
\(\displaystyle x_1= 1+ \frac{x_2+ 1}{2}= \frac{3+ x_2}{2}\)
\(\displaystyle x_2= 1+\frac{x_1+ x_3}{2}= \frac{ 2+ x_1+ x_3}{2}\)
\(\displaystyle x_3= 1+ \frac{x_2+ 5}{2}= \frac{7+x_2}{2}\).

Multiplying each equation by 2, we can reduct those to
\(\displaystyle 3- 2x_1+ x_2= 0\) or \(\displaystyle 2x_1- x_2= 3\)
\(\displaystyle 2+ x_1- 2x_2+ x_3= 0\) or \(\displaystyle -x_1+ 2x_2- x_3= 2\)
\(\displaystyle 7+ x_2- 2x_3= 0\) or \(\displaystyle -x_2+ 2x_3= 7\).

Three linear equations to solve for \(\displaystyle x_1\), \(\displaystyle x_2\), and \(\displaystyle x_3\).

 

[Argile1845]

Here is what I suggested:

You have written an expression for the sum that they ask for, but what you need is an equation that says what I put in bold above.

So, what is the average of the two expressions? Write an equal sign, followed by x, and then solve.

Mom here. Didn't let him see the answer yet
but I did explain to him the 1, x1, x2, x3, 5 placement and to think of it that way and that helped a lot. Here is his work so far.

Here is what I suggested:

You have written an expression for the sum that they ask for, but what you need is an equation that says what I put in bold above.

So, what is the average of the two expressions? Write an equal sign, followed by x, and then solve.

Oh I see. Thank you! I'll keep working on it.