Math Problem Statement

jawablah pertanyaan berikut

Suku ketiga dalam suatu barisan aritmetika adalah 11 dan suku kedelapan adalah 26. Carilah suku pertama dan beda dari barisan aritmetika tersebut! Carilah suku ke-n! Tentukan suku ke-20! Tentukan nilai n sehingga u_n = 152!

Solution

The problem you shared is about solving for terms in an arithmetic sequence (barisan aritmetika). Let's break down the problem:

Given Information:

  • The third term (U3U_3) of an arithmetic sequence is 1111.
  • The eighth term (U8U_8) is 2626.

Questions:

  1. Find the first term (aa) and the common difference (dd) of the sequence.
  2. Find the general formula for the nn-th term (UnU_n).
  3. Determine the 20th term (U20U_{20}).
  4. Find the value of nn such that Un=152U_n = 152.

Step-by-Step Solution:

1. Finding aa and dd

The general formula for the nn-th term of an arithmetic sequence is: Un=a+(n1)dU_n = a + (n-1)d For U3=11U_3 = 11, we have: 11=a+(31)d=a+2d11 = a + (3-1)d = a + 2d For U8=26U_8 = 26, we have: 26=a+(81)d=a+7d26 = a + (8-1)d = a + 7d

Now, solve this system of equations: a+2d=11(1)a + 2d = 11 \quad (1) a+7d=26(2)a + 7d = 26 \quad (2)

Subtract equation (1) from equation (2): (a+7d)(a+2d)=2611(a + 7d) - (a + 2d) = 26 - 11 5d=155d = 15 d=3d = 3

Now substitute d=3d = 3 into equation (1): a+2(3)=11a + 2(3) = 11 a+6=11a + 6 = 11 a=5a = 5

Thus, the first term a=5a = 5 and the common difference d=3d = 3.

2. General Formula for UnU_n

Now that we have a=5a = 5 and d=3d = 3, the general formula for the nn-th term is: Un=5+(n1)3=5+3n3=3n+2U_n = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2

3. Finding the 20th Term

Using the general formula Un=3n+2U_n = 3n + 2, we substitute n=20n = 20: U20=3(20)+2=60+2=62U_{20} = 3(20) + 2 = 60 + 2 = 62

Thus, the 20th term is 6262.

4. Finding nn such that Un=152U_n = 152

Set Un=152U_n = 152 in the general formula and solve for nn: 152=3n+2152 = 3n + 2 1522=3n152 - 2 = 3n 150=3n150 = 3n n=1503=50n = \frac{150}{3} = 50

Thus, n=50n = 50.

Final Answers:

  1. First term (aa) = 5, common difference (dd) = 3.
  2. General formula: Un=3n+2U_n = 3n + 2.
  3. 20th term (U20U_{20}) = 62.
  4. n=50n = 50 when Un=152U_n = 152.

Would you like any more details or have further questions?


Here are 5 related questions:

  1. How do you find the sum of the first nn terms of an arithmetic sequence?
  2. What happens to the arithmetic sequence if the common difference is negative?
  3. How would you determine if a sequence is arithmetic or geometric?
  4. Can an arithmetic sequence have non-integer terms? How?
  5. How do you find the common difference if you're only given non-consecutive terms?

Tip: In arithmetic sequences, the difference between consecutive terms is always constant, which makes it easier to identify and calculate other terms in the sequence.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Algebra

Formulas

U_n = a + (n-1)d

Theorems

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Suitable Grade Level

Grades 7-9