We've got a project where we're required to interpolate. We've got 10 values, square roots of 10 to 1. These roots are then adjusted by adding 50% to root of 10, 40% to root of 9, 30 % to root of 8, 20% to root of 7 and 10% to root of 6. Then again, 10% to root of 5, 20% to root of 4, 30% to root of 3, 40% to root of 2 and 50% to root of 1.

It looks like this:

3.16 + 50%=4.74

3 + 40%=4.2

2.83 + 30%=3.68

2.65 + 20%=3.17

2.45 + 10%=2.69

2.24 + 10%=2.46

2 + 20%=2.4

1.73 + 30%=2.25

1.41 + 40%=1.98

1 + 50%=1.5

Now we take cumulative totals and the final total is 29.08.

Now, if 10 is 4.74, 20 is 8.94(sum of 4.74 and 4.2)....90 is 27.58(sum of 1st 9 items or say 29.08-1.5)...

Can I interpolate and find values of say 3, 27, 67, 83 etc. now that we can values of all multiples of 10?

I actually tried using Newton's Approximation Formula but it either didn't work all the time or it seems it is giving approximate values which is of little use.