The ratio of two numbers is $2: 3$. when 6 is subtracted from the smaller of the two numbers and 7 is added to the larger of the two, the new ratio is $2: 7$. what was the smaller of the two numbers before 6 was subtracted from it

  14

Step-by-step explanation:

Let the smaller and larger numbers be represented by S and L, respectively.

  S/L = 2/3 . . . . the original ratio of the numbers

  3S = 2L . . . . . multiply by 3L

__

  (S-6)/(L+7) = 2/7 . . . . the new ratio of the numbers

  7(S-6) = 2(L+7) . . . . . multiply by 7(L+7)

  7S -42 = 2L +14

Using the expression for 2L from the first ratio, we can substitute to get ...

  7S -42 = 3S +14 . . . . substitute 3S for 2L

  4S = 56 . . . . . . . . . . .add 42-3S

  S = 14 . . . . . . . . . . . . . divide by 4

The original smaller number was 14.

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for