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Zorluk: OrtaArithmetic and Geometric Sequences and Series

A recipe calls for 12\frac{1}{2} cup of sugar on the first day of a fermentation process. Each day after that, the ratio of the sugar added on that day to the sugar added on the previous day is 1:31:3. What is the total amount of sugar, in cups, added during the first 3 days?

  1. 1318\frac{13}{18}Cevap
  2. B
    326\frac{3}{26}
  3. C
    29\frac{2}{9}
  4. D
    52\frac{5}{2}
  5. E
    2132\frac{21}{32}

Cevap

The total amount of sugar added is 1318\frac{13}{18} cups.
The correct answer is 1318\frac{13}{18}. The sugar added on Day 1 is 12\frac{1}{2} cup. Since the ratio between consecutive days is 1:31:3, the common ratio is r=13r = \frac{1}{3}. This gives a Day 2 amount of 12×13=16\frac{1}{2} \times \frac{1}{3} = \frac{1}{6} cup, and a Day 3 amount of 16×13=118\frac{1}{6} \times \frac{1}{3} = \frac{1}{18} cup. Summing these three amounts using a common denominator of 18 yields 918+318+118=1318\frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18} cups.

Adım Adım Çözüm

1
Identify the type of sequence and the first term.
The first term is a1=12a_1 = \frac{1}{2}. The sequence is geometric because the ratio between the amounts added on consecutive days is constant.
The problem states that the ratio of sugar added on consecutive days is 1:31:3, which establishes a common ratio for a geometric sequence.
2
Determine the common ratio and find the terms for the second and third days.
The common ratio is r=13r = \frac{1}{3}. The second term is a2=12×13=16a_2 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}, and the third term is a3=16×13=118a_3 = \frac{1}{6} \times \frac{1}{3} = \frac{1}{18}.
Each subsequent term of a geometric sequence is found by multiplying the previous term by the common ratio.
3
Calculate the sum of the first three terms.
The total sum is S3=12+16+118=918+318+118=1318S_3 = \frac{1}{2} + \frac{1}{6} + \frac{1}{18} = \frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18}.
Finding the total amount requires summing the individual amounts added over the three days using a common denominator of 18.

Anahtar Kavram

Sum of a finite geometric series

Alternatif Yöntem

Instead of calculating and adding the individual terms, you can use the sum of a finite geometric series formula: Sn=a1(1−rn)1−rS_n = \frac{a_1(1-r^n)}{1-r}. Substituting a1=12a_1 = \frac{1}{2}, r=13r = \frac{1}{3}, and n=3n=3 yields: S3=12(1−(13)3)1−13=12(1−127)23=12(2627)23=1327×32=1318S_3 = \frac{\frac{1}{2}\left(1 - \left(\frac{1}{3}\right)^3\right)}{1 - \frac{1}{3}} = \frac{\frac{1}{2}\left(1 - \frac{1}{27}\right)}{\frac{2}{3}} = \frac{\frac{1}{2}\left(\frac{26}{27}\right)}{\frac{2}{3}} = \frac{13}{27} \times \frac{3}{2} = \frac{13}{18}.
Tahmini Süre:1m 30s
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Arithmetic and Geometric Sequences and Series

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