The largest and smallest of three consecutive terms in an arithmetic sequence differ by 14. half of the smallest term is added to each term and the sum of the resulting three numbers is 120. what is the value of the original smallest term?

Let the smallest number be x and the common difference be d .

So the three consecutive numbers are

x , x+d, x+2d

And it is given that the difference between smallest and largest is 14.

So we get x+2d-x =14

2d=14

d=7

So the three consecutive numbers are

x , x+7, x+14 .

Half of the smallest number is 0.5x.

Adding it to each number will give x+0.5x, x+7+0.5x, x+14+0.5x

= 1.5x , 1.5x +7 , 1.5x +14

And there sum is 120 .

1.5x +1.5x +7 +1.5x+21 =120

4.5x +28=120

4.5x = 92

x= 20 approx .