Math Problem Statement

Tiga suku pertama deret geometri adalah a, 2a - 10, dan 4a + 10. Pernyataan yang benar adalah ....

Solution

The image contains a problem related to geometric sequences (deret geometri). The question appears to be:

"Tiga suku pertama deret geometri adalah aa, 2a102a - 10, dan 4a+104a + 10. Pernyataan yang benar adalah ...."

This translates to: "The first three terms of a geometric sequence are aa, 2a102a - 10, and 4a+104a + 10. Which of the following statements is true?"

Let's solve this problem step by step:

Step 1: Identify the formula for a geometric sequence

In a geometric sequence, the ratio between consecutive terms is constant. That is, the ratio rr between the second and first term is equal to the ratio between the third and second term:

r=Second termFirst term=Third termSecond termr = \frac{\text{Second term}}{\text{First term}} = \frac{\text{Third term}}{\text{Second term}}

Substitute the terms: r=2a10a=4a+102a10r = \frac{2a - 10}{a} = \frac{4a + 10}{2a - 10}

Step 2: Solve the equation

Now, we solve the equation:

2a10a=4a+102a10\frac{2a - 10}{a} = \frac{4a + 10}{2a - 10}

To solve this, cross multiply:

(2a10)(2a10)=a(4a+10)(2a - 10)(2a - 10) = a(4a + 10)

Expanding both sides:

(2a10)2=a(4a+10)(2a - 10)^2 = a(4a + 10)

4a240a+100=4a2+10a4a^2 - 40a + 100 = 4a^2 + 10a

Step 3: Simplify the equation

Cancel out the 4a24a^2 terms:

40a+100=10a-40a + 100 = 10a

Move all terms involving aa to one side:

40a10a=100-40a - 10a = -100

50a=100-50a = -100

Divide both sides by 50-50:

a=2a = 2

Step 4: Verify the solution

Substitute a=2a = 2 into the original terms:

  • First term: a=2a = 2
  • Second term: 2a10=2(2)10=410=62a - 10 = 2(2) - 10 = 4 - 10 = -6
  • Third term: 4a+10=4(2)+10=8+10=184a + 10 = 4(2) + 10 = 8 + 10 = 18

The first three terms of the geometric sequence are 2, -6, and 18. Now, verify the common ratio rr:

r=62=3,r=186=3r = \frac{-6}{2} = -3, \quad r = \frac{18}{-6} = -3

Since the common ratio is constant, the sequence is valid.

Step 5: Conclusion

The correct value of aa is 2, and the ratio rr is 3-3. Therefore, the statement that is correct is:

  • The second term is 6-6 (not 6 as indicated in the options).
  • The third term is 1818.
  • The ratio is 3-3.

Would you like more details or have any other questions? Here are some follow-up questions you might find interesting:

  1. What are the properties of a geometric sequence?
  2. How can you find the sum of the first nn terms of a geometric sequence?
  3. What is the significance of a common ratio in geometric sequences?
  4. Can a geometric sequence have a common ratio of zero?
  5. How does a geometric sequence differ from an arithmetic sequence?

Tip: In geometric sequences, always ensure the ratio between consecutive terms is constant before assuming a sequence is geometric.

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Math Problem Analysis

Mathematical Concepts

Geometric Sequences
Ratios

Formulas

r = Second Term / First Term = Third Term / Second Term

Theorems

Geometric Sequence Ratio Formula

Suitable Grade Level

Grades 9-12