Math Problem Statement

tentukan suku ke-10 dari barisan bilangan 90,45, 45

Tentukan suku ke-10 dari barisan bilangan 90, 45, 45/2, 45/4, 45/8, ...

Solution

The problem asks to find the 10th term in the sequence that starts with:

90,45,452,454,458,90, 45, \frac{45}{2}, \frac{45}{4}, \frac{45}{8}, \dots

Identifying the Sequence:

From the given terms, the pattern suggests that each term is halved successively after the second term:

  • 9090
  • 4545
  • 452\frac{45}{2}
  • 454\frac{45}{4}
  • 458\frac{45}{8}

This indicates that the common ratio between consecutive terms is 12\frac{1}{2}, forming a geometric sequence after the first two terms.

General Formula for the Sequence:

The first term (a1a_1) is 9090, and the second term (a2a_2) is 4545. The terms after the second follow the form of:

an=45×(12)n2forn2a_n = 45 \times \left(\frac{1}{2}\right)^{n-2} \quad \text{for} \, n \geq 2

Finding the 10th Term:

Now, we want to find the 10th term. For n=10n = 10:

a10=45×(12)102=45×(12)8a_{10} = 45 \times \left(\frac{1}{2}\right)^{10-2} = 45 \times \left(\frac{1}{2}\right)^8

a10=45×1256=45256a_{10} = 45 \times \frac{1}{256} = \frac{45}{256}

Thus, the 10th term of the sequence is 45256\frac{45}{256}.

Let me know if you need any further explanation or more details.


Related Questions:

  1. What is the 5th term in this sequence?
  2. How do you determine if a sequence is geometric?
  3. What is the sum of the first 10 terms in this geometric sequence?
  4. How does the common ratio affect the behavior of a geometric sequence?
  5. Can this sequence have a limit as nn approaches infinity?

Tip:

In geometric sequences, if the common ratio is less than 1, the terms approach 0 as nn increases.

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Math Problem Analysis

Mathematical Concepts

Geometric Sequence
Ratio

Formulas

General term of geometric sequence: a_n = a_1 * r^(n-1)
For this sequence: a_n = 45 * (1/2)^(n-2) for n >= 2

Theorems

Geometric progression with a common ratio less than 1 tends to zero

Suitable Grade Level

Grades 9-11