Sophie has 25 ounces of a 20% sugar syrup that she wants to dilute to a 10% sugar syrup. How much water does she need to add to obtain the 10% syrup?

If Sophie adds x oz of water, the volume of the new syrup will be (25 + x) oz. The syrup she starts with already contains 0.20 • (25 oz) = 5 oz of sugar, and if she adds water to it, that amount doesn't change.

She wants to end up with a syrup containing 10% sugar, so that the starting 5 oz makes of 10% of the total (25 + x) oz of syrup. So

5 / (25 + x) = 0.10

5 = 0.10 (25 + x)

5 = 2.5 + 0.10x

2.5 = 0.10x

x = 2.5 / 0.10

x = 25 oz

, 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