Lana follows a recipe to make muffins. She finds they are too sweet, so in the next batch, she uses three-fourths as much sugar. She continues refining the recipe by using three-fourths as much sugar in each successive batch. If she used 500 grams of sugar in the first batch, which expression can Lana use to find the total number of grams of sugar she will use to make the first 6 batches of muffins
500(-1(3/4)^6)
500(1-3/4)/1-3/4
500(-1(3/4)^6)/(1-3/4)
500(1-(3/4))^6/(1-3/4)
Lana can use the expression 500(1-(3/4))^6/(1-3/4) to find the total number of grams of sugar she will use to make the first 6 batches of muffins.
can you show your work on how you got this answer
Certainly!
Since Lana used 500 grams of sugar in the first batch and she reduces the amount by three-fourths in each successive batch, the amount of sugar she uses in each batch can be given by the geometric sequence:
500, (3/4) * 500, (3/4)^2 * 500, (3/4)^3 * 500, ...
In general, the nth term of this sequence can be calculated using the formula:
a_n = a_1 * r^(n-1)
where a_1 is the first term, r is the common ratio, and n is the number of terms.
From the given information, we know that a_1 = 500 and r = 3/4.
To find the total amount of sugar used in the first 6 batches, we need to find the sum of the first 6 terms of the sequence.
The formula to calculate the sum of a geometric series is:
S_n = a_1 * (1 - r^n) / (1 - r)
Plugging in the values, we have:
S_6 = 500 * (1 - (3/4)^6) / (1 - 3/4)
Simplifying the expression:
S_6 = 500 * (1 - 729/4096) / (1/4)
S_6 = 500 * (3667/4096) / (1/4)
S_6 = 500 * (3667/4096) * (4/1)
Simplifying further:
S_6 = 500 * 3667 / 1024
S_6 = 1,830,500 / 1024
S_6 ≈ 1786.914
Therefore, Lana will use approximately 1786.914 grams of sugar in the first 6 batches of muffins.
can you solve again without simplifying
Certainly! Let's solve it without simplifying the expression.
We have the geometric sequence for the amount of sugar used in each batch:
500, (3/4) * 500, (3/4)^2 * 500, (3/4)^3 * 500, ...
The expression for the sum of the first 6 terms of this sequence can be written as:
500(1 - (3/4)^6) / (1 - 3/4)
= 500(1 - 729/4096) / (1/4)
= 500(3667/4096) / (1/4)
= 500 * 3667/4096 * 4/1
= 500 * 14668/4096
= 500 * 3.57861328125
= 1789.306640625
Therefore, Lana will use approximately 1789.306640625 grams of sugar in the first 6 batches of muffins.