I have a list of five numbers, where the first number is 1 and the fifth number is 5. 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 sum of all five numbers in the list is 25.

According to the question,

The first number on my list of five numbers is 1, and the fifth is 5. The other figures are all one more than their two nearest neighbors’ averages.

Let the list of numbers be 1,x,y,z,5.

As each of the other numbers is equal to one more than the average of its two neighbors.

x=(1+y)/2+1

2x=3+y -(1)

Similarly,

y=(x+z)/2+1

2y=x+z+2 -(2)

z=(y+5)/2+1

2z=y+7 -(3)

Multiply equation (2) by 2 and then substitute equations (1) and (3),

sumof all five numbers in thelistis 25.listof five numbers is 1, and the fifth is 5. The other figures are all one more than their two nearest neighbors’ averages.listof numbers be 1,x,y,z,5.equation(2) by 2 and then substituteequations(1) and (3),equation(1),equation(3),listis: 1,5,7,7,5Sumof the numbers oflist= 1+5+7+7+5 = 25sumhere: