Lily and sara each had an equal amount of money at first. after lily
spent $18 and sara spent $25, lily had twice as much as sara. how
much money did each have at first?

32

Step-By-Step: First, Lily spent $18, so the amount of money Lily has now is x - 18. Sara spent $25, so likewise the amount of money Sara has now is x - 25. We know that Lily now has twice as much money as Sara, so we can write this fact algebraically as:
+x+-+18+=+2%28x+-+25%29+ (the amount of money Lily has now = two times the amount of money Sara has now)
Now we solve for x:
+x+-+18+=+2x+-+50+ (distribute the 2 on the right hand side)
+-18+=+x+-+50+ (subtract x from both sides)
+32+=+x+ (add 50 to both sides)
So we got x = 32. Does this make sense? If both Lily and Sara started with $32, Lily would now have $32 - $18 = $14, while Sara would now have $32 - $25 = $7, so Lily would now have twice as much money as Sara. Thus $32 is the answer.

, and

for

step-by-step explanation:

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

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

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