What is the sequence of these numbers 1,2,3,4 4,19,37,49

Question

I am given the numbers 1,2,3,4 followed by the numbers 4,19,37,49.  The fomula is  XtimesNumbergiven+some number will equal the second set of numbers.

Answer 1

If the numbers in the first set are related to those in the second set we can use the formula y_n=ax_n+b, where x_n belongs to the first set and the corresponding y_n belongs to the second set.

Therefore if n=1: y₁=ax₁+b, 4=a+b; 19=2a+b. We have two equations and 2 unknowns so we can solve for a and b:

(2a+b)-(a+b)=19-4, a=15.

a+b=4 so 15+b=4 and b=4-15=-11, so y_n=ax_n+b, y_n=15x_n-11.

If the formula is correct then y₃=15x₃-11=45-11=34 which does not match y₃=37. Also y₄=60-11=49, which does match.

This suggests that the second set should have been 4, 19, 34, 49. This is an arithmetic progression with common difference=15, the value of a.